Welcome and enjoy your trip. I am looking forward to getting out to see your campus and stadium in 2019 and getting another B1G road trip under my belt..
You already have some good stuff here. I'll try to offer some things that are not so obvious from looking around the internet.
If you are going to fly Newark Airport is much closer and convenient than the other two NYC area airports. It makes a difference
The portion of New Jersey between the airport and the campus is pretty congested routinely. With driving don't assume that miles traveled means much relative to travel time. When traffic backs up you can eat up a lot of time not going anywhere very fast.
I think you have two best options in overall travel plan. Stay in town (New Brunswick) or stay in Manhattan and "commute".
I think someone has already said don't be confused that the Stadium is in a town called Piscataway. It's right across the River from the historic heart of the Unviersity in a small city called New Brunswick. They are not far apart.
Overview map
The oldest, most historic, and traditional parts of campus are labeled Old Queens and Voorhees Mall on this
map.
There are lots of side trips possible and a thousands of hotel choices in nice towns in the vicinity. But if you want to immerse yourself in the campus and town, there are only two choices. The
Heldrich and
Hyatt. (There are decent hotels in Piscataway, some pretty close to the stadium, but basically you'll be staying in an office park, not recommended.) Heldrich and Hyatt are both in the middle of New Brunswick. Easy walk to the many eating and drinking places that are near the College Avenue campus. You can even get a get a shuttle t the game. It's not a bad walk about two miles. If you do want to see some of the more far flung areas of campus you can hop aboard a free campus bus system. You really wouldn't need a car once you are there assuming a short weekend trip. If you are driving both provide secure parking for a fee.
So, if you are not driving and don't want to rent a car how does it work? You can take a train from Newark Airport to New Brunswick.
NJ Transit provides regular service on the Northeast Corridor Line. Amtrak makes occasional stops.Then you can walk, shuttle, or bus, and if you want to pay taxi or uber wherever you need to be. Even pulling a piece of luggage it's an easy walk to either hotel from the station.
The other option is the NYC stay. If you are doing this the Newark Airport convenience is less important but still a consideration. If you don't get to this part of the country often and haven't seen Manhattan recently this is a very appealing option. There are even more hotel choices and I won't offer a suggestion beyond avoid anything not in Manhattan proper, preferably between midtown/Times Square and Downtown. Brooklyn and parts of Harlem might be hip but it's not where most tourists want to be. Avoid Queens, forget Bronx and Staten Island. If you are staying any of these you might as well stay in a nicer suburb in NJ close to the City. There are literally too many things to do in NYC to mention but you can find them on the net.
On game day you go to Penn Station via subway, cab or walk. Get on a
NJ Transit train for a 45 minute ride that takes you to New Brunswick and a walk or shuttle ride to the stadium. Look for Trains with a destination of "Trenton" or "Jersey Avenue". By a ticket from the vending machines in the station. You pay extra if the conductor has to sell you one on the train.
Let us know if you have any questions. When it gets closer we can talk about places to eat in New Brunswick. As to where to eat in Manhattan that's a whole other discussion.
Some other handy links
Visit Rutgers
Stadium Info including links to parking
New Brunswick City Market
