We are from nearby New Jersey and have spent many, many work and play days in Manhattan. There is no shortage of things to do in Mid-town (between 23rd ~ 72nd St. and Downtown (south of 14th St) Manhattan, or tours, guides and transport to facilitate the same.
I would agree that luggage storage is an issue (as it was for us when we dis-embarked at Dover, and toured Windsor, before boarding our flight at LHR), and Schwarz Travel & Luggage storage service
http://www.schwartztravel.com/ratesreserv.html
Seems like a good choice. Note that there are LOTS of transportation options to/from NYPennStation nearby, so this is a good "base" to get to first thing in the am, and then ~1 hour before one needs to be at the airport.
You really don't need to pay the Disney premium to "tour" NYC . . . because, you will never see more than a small fraction of NYC in 8 hrs, so my recommendation is to skip the various HonHoff buses, and
a) choose 1 place* or museum* you really want to see / experience, and take some time to enjoy it;
b) choose 1 place to dine or "food-tour," e.g, Chinatown, Jewish or Italian Deli, or a good steak place. Not always cheap, but typically fantastic. Please don't waste your time or stomach on a chain restaurant. The pizza is awesome.
Besides taxi's, Uber and Lyft vehicles are plentiful and 15~40% cheaper. The subways are usually faster. Taking the bus in Manhattan, well, except for short distances, well it just doesn't make sense. Btway, GoogleMaps provide some great insight when choosing transit.
Watch your valuables (like any big city), make sure you aren't late for your flight and enjoy your short visit!
*American Museum of Natural History, Metropolitan Museum of Art, Times Square (possibly a Off/Broadway Show if you have time on a Wednesday/Saturday/Sunday), Greenwich Village, the "Battery" and the view of the Harbor, Frick Collection (my favorite), Guggenheim, MusofModernArt, Central Park/Zoo, Rockefeller Center/30 Rock, are some typical choices. I am forgetting/ignoring twice as many other choices.