Cheap pass for a week in Tokyo and Nagoya?

Some additional info: JR Pass is about US$300. It now includes transport to Narita and Haneda airports. Tokyo has multiple JR (Japan Railways) branded train lines that are covered by the pass; however, Tokyo Metro and Toei Subways and busses are not covered. Last year, we found that it was easy enough to get to where we wanted by JR Pass. See http://www.jreast.co.jp/e/info/map_a4ol.pdf

There are regional JR passes, but they don't help for your specific case.

If you don't want to see Tokyo at all, you can simply fly. Mid-week round trip tickets are about $140 US.

Well, the JR Pass is actually probably worth the price if your flight is to and from Narita (NRT), but less so if it's to and from Haneda (HND).

Google tells me that a one-way ticket from Tokyo Station/Shinagawa Station to Nagoya Station via the Tokaido Shinkansen (東海道新幹線), on the Kodama or Hikari trains, costs 10,880 yen (source: here, in Japanese), or ~95 USD.

The price of the 7-day JR Pass is 29,110 yen (~254 USD). A roundtrip between Tokyo and Nagoya via the Tokaido Shinkansen costs 21,760 yen - so if you spend at least 7,350 yen riding on JR trains (other than the Shinkansen trip), your JR Pass would be worth the bucks. A ride on the JR East Narita Express (成田エクスプレス) train from Narita Airport to Shinjuku Sta. in Tokyo costs 3,190 yen, and thus 6,380 yen round trip. If you then spend at least 970 yen riding on other JR trains, your JR Pass will be worth the 29,110 yen.

For your reference, a ride from Akihabara Sta. to Shibuya Sta. (16km), for example, costs 200 yen. So, if you take more than 5 trips on JR trains in Tokyo (very likely), you would be saving money with the JR Pass.

But since Nagoya is your main destination, you will also need something else. While riding only on JR trains could get you basically everywhere in Tokyo, JR trains can't really get you anywhere in the city of Nagoya. You will need to take the Nagoya Subway (名古屋市営地下鉄), and, to a lesser extent, the Meitetsu (名鉄), or Nagoya Railroad.

There are day passes for the Nagoya Municipal Subway & Buses, at 850 yen per pass, which is not a bad deal (one subway ride costs 200+ yen). You could buy them at any Nagoya Subway station, or onboard Nagoya Municipal Buses (see here).

So, conclusion: if your flight is to and from Narita, I'd buy a 7-day JR Pass and use day passes for transportation within Nagoya. If your flight is to and from Haneda, I wouldn't get any pass. I would just buy a PASMO or Suica card (which I recommend you to do even if you buy a JR Pass for its convenience), use the card to ride subways and JR trains in both Tokyo and Nagoya, and buy tickets for the Tokaido Shinkansen at the ticket center in Ueno Station or Tokyo Station.

There is no one "pass", there are many passes that cover different areas and services.

As @jpatokal stated, there is no "magic cover all" pass. There are however many passes that can be purchased on the cheap that cover some / all services you intend to patronise.

TOKYO

Tokyo Metro 24-Hour Ticket
A:￥600       C:￥300
Allows unlimited travel for 24 hours on all Tokyo Metro lines.
Purchase at any Tokyo Metro Vending Machine on or before day of travel.

Common One-day Ticket for Tokyo Metro & Toei Subway
A:￥1000     C:￥500
Allows unlimited travel for 24 hours on all Tokyo Metro and Toei Subway lines.
Purchase at any Tokyo Metro or Toei Subway Vending Machines on or before day of travel.

Tokyo Combination Ticket
A:￥1590     C:￥800
Allows unlimited travel for one day on all lines of Tokyo Metro, Toei Subway, Toei Streetcar (Toden), Toei Bus (except special late-night buses and buses with seating capacity restrictions), all routes of Nippori-Toneri Liners, and all JR lines within the Tokyo metropolitan area.
Purchase at any Vending Machines / Ticket Window of the above mentioned companies on or before day of travel. Must be used within a month of purchase.

Keikyu Haneda/Subway Pass
A:￥1310     C:￥660
A set that includes a one-way ticket from Haneda Airport Domestic Terminal Station or International Terminal Station to Sengakuji Station on the Keikyu Line, and a Common One-day Ticket for Tokyo Metro & Toei Subway.
Purchase at Haneda Airport Domestic Terminal Station or International Terminal Station on the Keikyu Line on the day of travel. Not sold by either Tokyo Metro or Toei Subway.

WELCOME! Tokyo Subway Ticket
Subway 24hr + Keikyu One Way Ticket   A:￥1200     C:￥600
Subway 48hr + Keikyu One Way Ticket   A:￥1600     C:￥800
Subway 72hr + Keikyu One Way Ticket   A:￥1900     C:￥950
Subway 24hr + Keikyu Return Ticket       A:￥1500     C:￥750
Subway 48hr + Keikyu Return Ticket       A:￥1900     C:￥950
Subway 72hr + Keikyu Return Ticket       A:￥2200     C:￥1100
Allows either a one way trip to/from Haneda airport on the Keikyu line (one day of purchase) or a return trip to/from Haneda Airport on the Keikyu line (within 14 days of purchase) as well as unlimited travel for 24 / 48 / 72 hours on all Tokyo Metro and Toei Subway lines.
Purchase at any Tokyo Metro or Toei Subway Vending Machines on or before day of travel.

Limousine Bus & Subway Pass (Haneda Airport)
Subway 24hr + Limousine Bus One Way Ticket    A:￥1800     C:￥900
Subway 48hr + Limousine Bus Return Ticket        A:￥3200     C:￥1600
Subway 72hr + Limousine Bus Return Ticket        A:￥3500     C:￥1750
Allows either a one way trip or two one way trips on the Airport Limousine Bus between Haneda and downtown Tokyo within 6 months of purchase and unlimited travel for 24 / 48 / 72 hours on all Tokyo Metro and Toei Subway lines.
Purchase at Information Counter at Haneda Airport Arrival Lobby, Limousine Bus ticket counters at Shinjuku Station West Exit and Tokyo City Air Terminal (TCAT) 3F, and Tokyo Metro pass offices (except at Nakano, Nishi-funabashi and Shibuya (Fukutoshin Line) Stations).

Keisei Skyliner & Tokyo Subway Ticket
Subway 24hr + Keisei Skyliner One Way Ticket    A:￥2800     C:￥1400
Subway 48hr + Keisei Skyliner One Way Ticket    A:￥3200     C:￥1600
Subway 72hr + Keisei Skyliner One Way Ticket    A:￥3500     C:￥1750
Subway 24hr + Keisei Skyliner Return Ticket        A:￥4700     C:￥2350
Subway 48hr + Keisei Skyliner Return Ticket        A:￥5100     C:￥2550
Subway 72hr + Keisei Skyliner Return Ticket        A:￥5400     C:￥2700
Allows either a one way trip to/from Narita Airport (either terminal) and Keisei Ueno Station on the Keisei Skyliner line (one day of purchase) or a return trip to/from Narita Airport (either terminal) and Keisei Ueno Station on the Keisei Skyliner line as well as unlimited travel for 24 / 48 / 72 hours on all Tokyo Metro and Toei Subway lines.
Purchase at Narita Airport Station or Airport Terminal 2 Station, Keisei Electric Railway ticket counters at Narita Airport Arrival Lobby (Only sold to visitors from overseas (passports will be checked)), onboard domestic Peach Aviation and Spring Airlines flights bound for Narita Airport.

Limousine Bus & Subway Pass (Narita Airport)
Subway 24hr + Limousine Bus One Way Ticket    A:￥3400     C:￥1700
Subway 48hr + Limousine Bus Return Ticket        A:￥3700     C:￥2850
Subway 72hr + Limousine Bus Return Ticket        A:￥6000     C:￥3000
Allows either a one way trip or two one way trips on the Airport Limousine Bus between Narita and downtown Tokyo within 6 months of purchase and unlimited travel for 24 / 48 / 72 hours on all Tokyo Metro and Toei Subway lines.
Purchase at Limousine Bus ticket counters at Narita Airport Arrival Lobby, Shinjuku Station West Exit and Tokyo City Air Terminal (TCAT) 3F, and Tokyo Metro pass offices (except at Nakano, Nishi-funabashi and Shibuya (Fukutoshin Line) Stations).

Tokyo Shuttle & Subway Pass
Subway 24hr + Tokyo Shuttle One Way Ticket    A:￥1700
Subway 48hr + Tokyo Shuttle One Way Ticket    A:￥2100
Subway 72hr + Tokyo Shuttle One Way Ticket    A:￥2400
Subway 24hr + Tokyo Shuttle Return Ticket        A:￥2600
Subway 48hr + Tokyo Shuttle Return Ticket        A:￥3000
Subway 72hr + Tokyo Shuttle Return Ticket        A:￥3300
Allows either a one way trip (valid for day of purchase) or a return trip (valid within 8 days) on the Tokyo Shuttle between Narita Airport and Central Tokyo and unlimited travel for 24 / 48 / 72 hours on all Tokyo Metro and Toei Subway lines.
Purchase at either Keisei Bus Ticket Counter or Narita Airport Arrival Lobby.

Odakyu Tokyo Metro Pass
A:￥770-1880     C:￥390-940
Allows one round trip between the station of departure on Odakyu lines and Yoyogi-uehara Station and unlimited travel for one day between Yoyogi-uehara Station of the Odakyu line and Shinjuku Station and on all Tokyo Metro lines.
Purchase at any Odakyu or Tokyo Metro Vending Machine on same day of travel or any Odakyu or Tokyo Metro Ticket Counters for same day or before travel.

Similar tickets exist to the Odakyu combination ticket, for most rail companies servicing Tokyo. (Others include: Tobu Tokyo Metro Pass, Tojo Tokyo Metro Pass, Tokyu Tokyo Metro Pass, Toyo Tokyo Metro Pass, SR Tokyo Metro Pass, TX Tokyo Metro Pass, Seibu Tokyo Metro Pass, Tokyu Tobu Tojo Hiking Ticket, Tokyu Seibu Marugoto Ticket, Tokyu Tobu Tojo Koedo Kawagoe Coupon, Seibu Yokohama Bayside Ticket, Seibu Tokyu Triangle Ticket, Tojo Yokohama Bayside Ticket, Tojo Tokyu Triangle Ticket).

NAGOYA

Donichi Eco Kippu
A:￥600     C:￥300
Allows unlimited travel on Nagoya Subway Lines, Nagoya City Bus Routes and the Sightseeing Bus Me-guru Route for Saturdays, Sundays, Public Holidays and the 8th day of every month on the day of purchase as well as discounts at specified tourist locations.
Purchase at the iCenter at Oasis 21, the Kanayama Tourist Information Center, Nagoya Subway ticket counters, aboard city buses, stationmaster offices, city bus offices and Transportation Bureau Service Centers (commuter pass sales counters).

Bus & Subway ticket
A:￥850     C:￥430
Allows unlimited travel on Nagoya Subway Lines, Nagoya City Bus Routes and the Sightseeing Bus Me-guru Route for any day of the week on the day of purchase as well as discounts at specified tourist locations.
Purchase at the iCenter at Oasis 21, the Kanayama Tourist Information Center, Nagoya Subway ticket counters, aboard city buses, stationmaster offices, city bus offices and Transportation Bureau Service Centers (commuter pass sales counters).

Subway Ticket
A:￥740     C:￥370
Allows unlimited travel on Nagoya Subway Lines for any day of the week on the day of purchase as well as discounts at specified tourist locations.
Purchase at the iCenter at Oasis 21, the Kanayama Tourist Information Center, Nagoya Subway ticket counters, aboard city buses, stationmaster offices, city bus offices and Transportation Bureau Service Centers (commuter pass sales counters).

Bus Ticket
A:￥600     C:￥300
Allows unlimited travel on Nagoya City Bus Routes and the Sightseeing Bus Me-guru Route for any day of the week on the day of purchase as well as discounts at specified tourist locations.
Purchase at the iCenter at Oasis 21, the Kanayama Tourist Information Center, Nagoya Subway ticket counters, aboard city buses, stationmaster offices, city bus offices and Transportation Bureau Service Centers (commuter pass sales counters).

Sightseeing Bus Me-guru
A:￥500     C:￥250
Allows unlimited travel on the Sightseeing Bus Me-guru Route for any day of the week on the day of purchase as well as discounts at specified tourist locations.
Purchase at the iCenter at Oasis 21, the Kanayama Tourist Information Center, Nagoya Subway ticket counters, aboard city buses, stationmaster offices, city bus offices and Transportation Bureau Service Centers (commuter pass sales counters).

There are numerous other passes available depending on the company/companies involved, the location of the services and the time of day/day you intend to utilise the service. This is a list of most of the major passes available in Tokyo and Nagoya and may not include all options. Asking at a tourist / Information desk will usually get you up to date information as to which passes are currently sold and which passes are right for your circumstances.