We were at the Menger just last weekend. It's currently being painted but should be finished by the time you go. The old section is lovely and has some fascinating ghost stories associated with it.
Fun things to do in San Antonio? There's a lot to do even if you don't have a car since there's a lot of attractions downtown. Explore the Riverwalk, visit the Alamo, go to the IMAX which has several movie choices including a great film about the Alamo. A lot of people enjoy the Buckhorn Saloon & Museum and the Institute of Texan Cultures among other museums. Don't bother with the Hertzberg Circus Collection which unfortunately has recently been shut down.
I like old churches so I enjoy walking to the old San Fernando Cathedral and then visiting the Spanish Governor's Palace on the same walking tour.
You can drive along the mission trail and tour the old missions if you have a car. There's also a Sea World, Splashtown and a Six Flags park in the area. A short drive (actually in New Braunfels) gets you to the best waterpark anywhere in my opinion, the Schlitterbahn. Gruene, which is now part of New Braunfels, is a neat old town with lots of shopping and exploring to do.
The Menger Hotel is right next to the Alamo and Rivercenter. Just across from the hotel is an entrance to the Riverwalk which goes through the Hyatt. There's also a Ripley's/Wax Museum just across from you and just down the street a new Ripley's haunted house which no one was willing to try with me.
As for food, there's many places along the Riverwalk. I haven't been for a while but Boudro's used to be good and County Line is good for barbecue. You can hop on a trolley and ride to Market Square for 50 cents. My husband loves Mi Tierra, a San Antonio institution but I personally prefer La Margarita for Tex Mex.
Good luck! You may want a guide book.
