Check this out: Official New England Patriots - Game Reports - Stadium Details
Now those times are for a typical 1:00 pm game. From what I have read, for Monday night games the train departs about a half hour after the game is finished and arrives about an hour before the start of the game. I would call them to check what their schedule will be like on that day - call 617-222-3200, or the ticket office info line at (800) 543-1776.
As far as train or renting a car, there are pros and cons to each:
Train: less expensive than cost of parking and renting a car; no sitting in traffic (and yes, you will be sitting in traffic for a long time); no concerns with drunk driving or driving on unfamiliar roads.
Car rental: if you want to tailgate - whether it be to barbecue, drink, or simply enjoy the pre-game atmosphere for a few hours - then you are going to miss all that if you take the train; no coolers, no drinking on the train.
As far as public transportation goes, what the US offers is sadly far behind what you are probably used to. For those going to the game, the only pre-game party is pretty much the one that is going on in the parking lots outside the stadium. In this case though you are in luck since it is a night game, so you may have a few options.
My question to you is this: where abouts are you staying? Once we know that we can give you some ideas on where you may want to go before the game.