Getting to Southampton in the Summer between the hours of ~(6:30-9:30AM) and out of Southampton from ~(3:30-6:30PM) is as awful of a travel expense as I could wish on someone. There's basically one road into town, CR-39/Montauk Highway NY-27, which chokes down to a single lane once you enter the Town of Southampton. I imagine on U.S Open weekend especially with the course right off the main drag it will be even worse. I'd do a quick google search to see if people have posted about what the traffic was like last time the Open was at Shinnecock Hills, but I would be surprised if it wasn't terrible.
So that being said, I think determining how much of an issue it will be to get into/out of town is first priority. If it turns out it's not a terrible situation, then I'd probably recommend somewhere on the North Fork. The beer scene on the South Fork is pretty small (aside from Southampton Publick House) while the North Fork has several breweries (Long Ireland, Moustache, Crooked Ladder, Jamesport Farm, Greenport Harbor). Those locations are probably spread out from west-to-east by about 20 miles, so transportation is a definite must. The North Fork is probably at least a 25-30min trip under ideal conditions to Shinnecock Hills, even longer the further east you go because you either have to loop around to get to the South Fork or take 2 ferries through Shelter Island.
I can't speak for Uber in Suffolk County Eastern LI. It's here. It hasn't been here for too long, but it does exist. I know there were some logistical hurdles in East Hampton, the Town east of Southampton, but I believe they have been all cleared up at this point.
I think staying in Southampton proper would be the easiest answer for getting to-and-from the U.S Open conveniently, but there's not much beer going on as compared to the North Fork, although there are plenty of great wineries if you enjoy wine. I'd expect it to be more expensive staying in Southampton, especially on U.S Open weekend because it's a pretty high society town in the Summer to begin with and there aren't really any hotel chain stye hotels. The North Fork has access to more chain hotels but they too may be pricey because it's becoming a Summer weekend destination to come out here and go to the wineries, farms, etc.
I know I just gave you really general information, but I hope it gives you an idea. Any more questions feel free to ask.