Upvote:2
Unfortunately access rules to a airline lounges are rather complicated and often not very clear.
United does sells one-time passes. Here are some of the rules
A one-time pass admits one person, with no guests, to a United Club location. Children under 2 years old may accompany someone accessing a United Club location with a one-time pass. A one-time pass cannot be used at United Arrivals Lounges, United Polaris lounges or Partner Lounges. Effective November 1, 2019, one-time pass holders will need to provide a same-day boarding pass for travel on United, Star Alliance or a contracted partner operated flight for entry into all United Club locations.
Access to Lounges may be denied for one-time pass users during busy times when Lounges are at or near capacity.
"Same day boarding pass" implies that you can use the lounge after arrival, but I have seen lounge agent interpreting this as "only before departure". Please also note that they can refuse entry if they feel it's too full. They don't appear to have a restriction on time.
You won't get into the Polaris lounge but there are 4 other United Clubs in Chicago that you can try. The clubs tend to have limited food and drink options and no showers as compared the Polaris lounge and other international business class lounges.
Finn Air appears to be using the Swissport lounge in Terminal 5. Terminals 1-3 are connected airside, but Terminal 5 isn't (as as I know), once you are out of United/American territory, there is no way back.
You can check lounges and access conditions for example at https://www.loungebuddy.com/ORD?source=homepage&filters=%7B%22tempClosed.0%22%3A%22open%22%7D