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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.
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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.
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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.