Arriving early by cab or Uber is a smart move as (as someone above pointed out) the train before the game is inconsistent, often arriving just before, just as, or just after, the game starts, leaving no time for tailgating or even enjoying a walk around Gillette stadium.
May I suggest, however, that you think about taking the train back to downtown Boston (South Station) as it will be difficult for a cab or Uber to pick you up after the game, when (if the game has been close and everybody stays until the end) the traffic jam is unbelievable.
Just this past week I did see signs after the game for Uber directing folks to walk south of the stadium past the Bass Pro Shop, then I don't know where to meet. You were still looking at lots of traffic doing this. You would need to buy your ticket in advance, as there is no station at the game. Not sure if you would have to pay for round-trip and eat the cost of the trip in.
Can you rent your own car for the day? That might be cheaper in the end, and you can arrive early, tailgate with the best of them, then get back to your hotel at your leisure. If you do this, however, do NOT park in a stadium lot (unless you plan to tailgate after the game. It takes hours to get out, lots of times fighting with drunks who are trying to merge into that same small spot as you). Go to one on Rte. 1 and make sure you are either able to make a right turn going in the direction you want AFTER the game, or go to a lot that advertises "EASY OUT" with a cop to stop traffic so you can take a left.
If you are staying in Boston, you want to park north of the stadium, and be able to turn right (same side of Rte. 1 as Gillette) when you leave, or left (opposite side of Rte. 1 as Gillette) with a state cop that stops the traffic and allows folks to turn out of the lot.
If you are coming real early (for a one o'clock game leave at 9:00 a.m.; for a 4:15 game leave at least 4 hours before, if not earlier) your driver (or you) should be okay taking you down Route 93 South to 95 South to Rte. 1 to the stadium. See how close the state cops will allow you to get to the Patriots Hall of Fame and the Pro Shop if you are getting dropped off.
If you are driving yourself, try parking at Lot 33 on Rte. 1 south ($50), my favorite, and a short walk down Rte. 1 that leaves you on the correct side of the stadium (the north side) to enjoy the Patriots Hall of Fame, the Pro Shop and Patriots Place. You can turn left out of the lot onto Rte. 1 north after the game with the assistance of the state cop stationed there.
Another idea after the game instead of tailgating (if you want to let the traffic die down whether you get a ride or not) is to grab a meal at one of the restaurants at Patriots Place. Try Tavolino (Italian) which does not have a long line if you buzz over there immediately after the game ends.
Sorry for the long-winded response. I have been going to games for many years and have my parking & arrival/departure routines down to a science. Gillette traffic and parking issues can be very overwhelming to the uninitiated. Just ask the Texans.
