@anon2021: In regard to things to do within the 5 mile walking privilege area, I am certainly no expert. However, I am familiar with the West Point Bowling Center you inquired about, which is just inside the gates. It is a really nice 10 lane facility, is open to the public for open bowling most nights til 9 p.m. (Friday & Saturday til 11), and is great for families and actually has decent food.
Just a short walk from there across the street I highly recommend the beautiful Thayer Hotel, which is perched so majestically above the Hudson. If you have a little time to hang out and want to grab some good food outside with heavenly views of the Hudson and vistas to the east, check out the Zulu Time Rooftop Bar and Lounge (rated 4 star on TripAdvisor). It is a nice outdoor, spacious and hip area for couples and families with good food and service - and those Heavenly views! If you want to eat inside, there are several other great options for that at The Thayer.
Once you leave there it is a short walk to another spot very special to my Plebe's Mom and I, the West Point visitors Center. Since you are walking in on ground level there is no way to be prepared for the inspiring, expansive sweeping vistas of the Hudson River Valley the minute you step inside. The huge towering wall of floor to ceiling windows just draws you to the edge for that all encompassing view. Once you have soaked that in it only gets better. The exhibits that wind through the Center are 1st rate, just so well done. It is a great, visual way to learn everything about West Point, from what to expect each year as a student to the heroes who shaped history that graduated from there. Of course after that, many will be inspired to shop for West Point gear at the Gift Shop. My wife and I have braved the throngs several different weekends and managed to spend hundreds of dollars on everything from clothes to West Point license plate frames.
Across from the Visitor Center is the West Point Museum. I have not been there, but from the outside it is a beautiful building, and if it as well done as the Visitor's Center I would imagine it is a good place to learn more about West Point's heritage.
As far as restaurants, we have only been to a few there in Highland Falls. La Hacienda was very good for Mexican food in a lively and well decorated building. Schade's, across from the Visitor's Center had a huge selection of many types of food, which was good as was the service. Also, there at West Point, I believe there are at least three Starbucks, including one in the library and one at Grant Hall. The one I know which is open to the public is a nice walk down from Washington Rd off Buckner Loop, and a very convenient location on campus. We have already spent many hours there, as it is one of those that is spacious with lots of comfortable seating.
Another nice option to the south, which according to what I can tell is within the 5 mile area ("grounds of Bear Mountain Hotel and adjoining state park facilities when traveling by vehicle"), is Bear Mountain. It appears to have good hiking trails - the Appalachian Trail goes right through it, a zoo, paddle boats for rent, picnicking in summer, a huge swimming pool, ice skating in winter and a playground with a famous carousel with 42 hand carved animals you can ride that are replicas of bears rabbits, foxes etc. In addition to picnics there are restaurant options, including Restaurant 1915 at the Bear Mountain Inn.
To the north it appears that the West Point Golf Course is within the defined area. When we have driven by it does proudly proclaim it is open to the public. Also, the Victor Constant Ski Area (located near the West Point U.S. Mint) appears to be within the 5 mile area, and it is open to the public offering downhill, snowboarding, X-Country, snowshoeing, a Snow Sport School, and lessons for all ages.
Back in the central core of the campus you can walk to the elegant Eisenhower Hall Theater, which is a great place to catch a show, whether it's Broadway, dancing, singing or a magic act, without having to go to New York City. They have shows every week or two from fall to spring.
All academic year long you can also walk to an NCAA Division 1 sports venue, whether it's Michie Stadium for a football game, the Tate Rink for hockey, Abner Doubleday (class of 1842) Field to watch the 2018 Patriot League champions play high caliber baseball, or any of the other myriad of sports that are contested there.
I promise you, that only scratches the surface, and if and when you ever possibly run out of things to do, catch the train across the river and take the hour ride to Grand Central Station in New York City. Stay inside and go immediately over to the Grand Central Oyster Bar and Restaurant for great food and drinks. Reflect on how inspired you were to have just left West Point. That's what my wife and I just did several weeks ago. We ate there and were thinking and talking about how proud we were of our DD, knowing she is at a place that will challenge her to the max. And if God-willing she makes it through she will have earned something priceless that will stay with her for the rest of her life.