How can we be sure what the first day of the month was 2000 years ago and how can we determine which years were leap years?

The simple answer is, we cannot be sure for the Jewish calendar because it depended on the sighting of the new moon with which weather conditions regularly interfered. There is also a dispute about whether the first of Nissan was calculated from the barley harvest or the equinox.

For example, here is one way to do the calculation, but not all will agree.

Jesus' Baptism

The beginning of Jesus’ ministry was marked by His baptism and anointing by the Holy Spirit, which, according to Luke 3:1 occurred in the 15th year of the reign of Tiberius Caesar. Unlike the Persian calendar, the Jews in New Testament times used the non-accession year dating, or inclusive reckoning. This means that when a monarch ascended the throne, the first incomplete calendar year (no matter how small) was counted as the first year. Since Tiberius ascended the imperial throne on 19 August 14 AD, his “fifteenth year” would be the twelve months beginning 1 Ethanim (September/October) 27 AD by the non-accession reckoning of the Jews. Since Jesus’ ministry lasted 3½ years according to the Gospel record (see below for this date), Jesus’ Baptism occurred about October/November, 27 AD.

There is further evidence of this date. Luke also records that Jesus was baptized when He was “about 30 years old” (Luke 3:23). While the exact date of Jesus’ birth is unknown, there is a narrow range from which to choose. It was after the Caesar Augustus’ census of 8 BC which took several years to complete. And, it was before Herod’s death in April 4 BC. Since shepherds were in the fields, Jesus must have been born before Nov 5 BC and probably about Sep/Oct 5 BC. In Oct 27 AD, He would have been 31 years old, or “about 30 years old”. Later dates suggested by some would stretch the meaning of Luke’s age too far.

Jesus' Crucifixion

Jesus’ death on 14th of Nissan. The usual dating systems require finding a year in the range 30 AD to 33 AD in which the Jewish Passover fell on Friday. Since we do not have definitive and precise information about the Jewish calendar (specifically the occurrence of “leap years” with the second Adar), such methods depend more on what is unknown than known. Even astronomical methods cannot help because they depend upon local meteorological conditions such as cloud obscuring a new moon. However, the overwhelming consensus from such methods, despite their limitations suggests that the crucifixion must have occurred during the Passover in Nisan (March/April), either in March, 30 AD, or, April, 31 AD. The former is too early as already shown (Jesus’ ministry lasted more than 2½ years), but the second accords with the known facts.

Despite the insuperable difficulties in knowing the local calendar of 2000 years ago, there are still a very limited range of possible dates from which to choose. If we accept that Jesus died on the afternoon of Friday, 14th of Nisan, then the first of Nisan must be at sunset on Friday, about 18:00 local time. In the years 30-33, there are only two years that are even possible, namely,

• 30 AD when New Moon occurs at 20:07, Wed, Mar 22 local time. Thus, Nisan 1 would begin about 1 day and 22 hours later on Mar 24. This accords with the Babylonian calendar (not used in Jerusalem) but such an event is almost too soon in date and too soon after sunset to be reliably seen.
• 31 AD when New Moon occurs at 13:53, Tues, Apr 10 local time. Thus, Nisan 1 would begin about 3 days and 4 hours later on Apr 17. This is more credible.

...the bottom line is he was "about" 30 when he was baptized. Could that mean he was 29 going on 30 or he was just turned 31?

Lk 3:23 means Jesus was from 26 to 34 years old at the time of his Baptism.

Now, how long did his ministry last?

It depends on the dates of his Baptism and Crucifixion. Hypothesized lengths range from little more than 1 year (Jan 29 - Apr 30) to little more than 5 years (Jan 28 - Apr 33).

Focusing on the dating of the beginning of the ministry of John the Baptist "in the fifteenth year of the hegemonias of Tiberius Caesar" (Lk 3:1), there are several dating methods that Luke may have followed according to the following tree of choices:

1st choice: type of regnal years:

• D. factual a.k.a. Dynastic regnal years,

• C. Calendrical regnal years.

Within D: event marking the beginning of Tiberius' reign:

• D.13. Granting of proconsular power co-extensive with Augustus', i.e. including Rome, on 27 June 13,

• D.14. Augustus' death on 19 August 14.

Within C: 3 independent binary choices:

• between the event marking the beginning of Tiberius' reign (as in D),

• between accession year and non-accession year (aka inclusive) reckoning,

• between the Syro-Macedonian and the Jewish calendar,

plus an additional binary choice in the case of the Jewish calendar:

• between the year beginning on Tishrei 1 as used by the Diaspora and the year beginning on Nisan 1 as used in the Second Temple.

Therefore there are 14 possible combinations in total, 2 within branch D and 12 within branch C, the latter group comprising 4 under the Syro-Macedonian calendar and 8 under the Jewish calendar. Of those 14 combinations:

• 2 are compatible with Jesus' Baptism in January 27,

• 8 are compatible with Jesus' Baptism in January 28 (they include the above 2),

• 13 are compatible with Jesus' Baptism in January 29 (they include the above 8), and

• 1 is compatible only with Jesus' Baptism in January 30, which is discarded.

how do we determine the year Jesus died if we know he died on a Friday, Nisan 14 and was raised from the dead Sunday, Nisan 16?

By looking at the dates of the pre-Nisan new moons. But first I have to broaden your question, because it is assuming Johannine cronology of the Passion, which is not the only possibility (though it is the one I hold):

According to - Jesus died - So that Nisan 1
               on Friday  - would have been a

the Synoptics - Nisan 15 - Friday

John's gospel - Nisan 14 - Saturday

Before the IV century CE the Jewish calendar was observational, with a court proclaiming the 1st day of each month (Rosh Chodesh) on the basis of witnesses reporting the sighting of the new crescent moon. Specifically, in the morning of the (provisionally) day 30 of the ending month the court assembled and waited for witnesses. If during the day the required number of qualifying witnesses testified to having seen the new crescent shortly after the previous sunset, then the court proclaimed that the ongoing day was in fact the 1st day of the new month. If no witness appeared then the ongoing day was confirmed as day 30 of the ending month and the next day was proclaimed as the 1st day of the new month without necessity of witnesses, since a lunar month can have only 29 or 30 days.

Since astronomical computations determine the date and time of the new moon (the conjunction) but the beginning of the month was decided on the basis of the first sighting of the new crescent, we have to link new moon with first visibility, which is an active field of research because the islamic calendar is still observational. I will bring into consideration only two pieces of information which are relevant for the purposes of this answer.

First, "the 'best time' for the first visibility of the thin crescent moon is directly proportional to site elevation", so that "at a site elevation of 1000 m or less [such as in Jerusalem, which is at 754 m above sea level] the crescent could be first seen 5 to 10 minutes after sunset" [1].

Thus, using from now on Israel Standard Time (IST = GMT + 2), we have for Jerusalem in the interval of interest, taking sunset times from [2]:

Day:                         Mar 6  Mar 20 Apr 4

Jerusalem sunset time (IST): 17:40  17:50  18:00

Best time for 1st sighting:  17:50  18:00  18:10

This is confirmed by the times of the first sightings in [3].

Second, the historical record for the shortest time between new moon and first sighting with the naked eye is 15.0 hours [4]. To note, this sighting did not happen by chance but as a result of an specific expedition for that purpose by a party of six amateur astronomers. Quoting from [4]:

John Pierce [...] has had previous experience at crescent watching. [...] The observers pre-calculated the position of the Moon (with respect to the sunset point) for a specific time so that they knew exactly where to look. Pierce spotted the Moon with the unaided eye, and the sighting was confirmed with a 12.5-inch telescope as seen by four people. Jan Kemp also saw the Moon with aneided vision, Clint Bach and Ed Byrd saw the crescent only through binoculars and the telescope, while Jim Golden and Travis Byrd could not see the Moon at all.

Therefore if a new moon occurred during Julian/Gregorian date N (delimited in Roman day reckoning, from midnight to midnight), then:

• The most probable time of 1st sighting of the new crescent is shortly after sunset of Julian/Gregorian date N+1.

• A 1st sigthing shortly after sunset of the same day N is theoretically possible, but extremely unlikely, if the new moon occurred before 03:00 IST (in March). This is so because the historical record for the shortest time between new moon and first sighting with the naked eye is 15.0 hours [4].

• A 1st sigthing shortly after sunset of day N+2 is possible if the new moon occurred after 12:00 IST (in March), given that moon ages at first sigthings can be over 50 hours [4]. The probability of such late sighting increases as the new moon time gets closer to 24:00.

With the above in mind, let's examine the pre-Nisan new moons of 29-34, taking new moon data from [5] (adding 02:00 to convert GMT to IST) and weekdays from [6]:

Year  Pre-Nisan - Most/Next probable    - Consequent Nisan 1
      new moon  - new crescent sighting - time span and weekday
      (IST)       (IST)

29   Apr 02 19:30 - Apr 03 18:10 - Apr 03 sunset - Apr 04 sunset, Monday

     late sighting  Apr 04 18:10 - Apr 04 sunset - Apr 05 sunset, Tuesday

30   Mar 22 19:47 - Mar 23 18:00 - Mar 23 sunset - Mar 24 sunset, Friday

     late sighting  Mar 24 18:00 - Mar 24 sunset - Mar 25 sunset, Saturday

31   Mar 12 00:20 - Mar 13 17:55 - Mar 13 sunset - Mar 14 sunset, Wednesday

     early sighting Mar 12 17:55 - Mar 12 sunset - Mar 13 sunset, Tuesday

31e  Apr 10 13:33 - Apr 11 18:15 - Apr 11 sunset - Apr 12 sunset, Thursday
                    
32   Mar 29 22:01 - Mar 30 18:05 - Mar 30 sunset - Mar 31 sunset, Monday

     late sighting  Mar 31 18:05 - Mar 31 sunset - Apr 01 sunset, Tuesday

33   Mar 19 12:39 - Mar 20 18:00 - Mar 20 sunset - Mar 21 sunset, Saturday

34   Mar  9 05:27 - Mar 10 17:55 - Mar 10 sunset - Mar 11 sunset, Thursday

34e  Apr  7 13:43 - Apr 08 18:10 - Apr 08 sunset - Apr 09 sunset, Friday

The entries YYe cover the (IMO unlikely) cases in which the ending year was embolismic.

Thus, the open possibilities are:

In year:  Jesus' could have been crucified according to the date in:

  30    either the Synoptics or John's Gospel
  33    John's Gospel
  34    the Synoptics

References

[1] Sultan, Abdul Haq. "'Best time'for the visibility of the lunar crescent." The Observatory, Vol. 126, pp. 115-118 (2006). https://adsabs.harvard.edu/full/2006Obs...126..115S

[2] https://www.timeanddate.com/sun/israel/jerusalem

[3] Hoffman, Roy E. "Observing the new Moon." Monthly Notices of the Royal Astronomical Society, Vol. 340, Issue 3, pp. 1039–1051 (2003). https://doi.org/10.1046/j.1365-8711.2003.06382.x

[4] Schaefer, Bradley E. "Lunar crescent visibility." Quarterly Journal of the Royal Astronomical Society, Vol. 37, p. 759 (1996). https://adsabs.harvard.edu/full/1996QJRAS..37..759S

[5] Fred Espenak, "Six Millennium Catalog of Phases of the Moon". 1 to 100 AD: http://astropixels.com/ephemeris/phasescat/phases0001.html

[6] https://www.fourmilab.ch/documents/calendar/