Do we actually have any exclusively ukulele shops here? I don't personally know of any, but I bet that Easy Music Center on King Street has ukes. As for live music, someone else will have to answer that one for you too. There are definitly live music spots, it just depends on when you go.
But for secret adventures on Oahu that tourists probably don't know about, go rock climbing. If you google Climb Aloha, and go to their site, you can probably arrange for a rock climbing session on the cliffs at Makapu'u point. If you haven't rock climbed in your life, it will be absolutly amazing. From the top of the wall at Makapu'u, you can look down like 400 feet and see the waves crashing at the bottom of the cliff...for real...not to mention miles of coastline. Rock climbing. Do it...but don't try it on your own, since you will NOTbe able to find the hidden rock face on your own, and probably want equiptment.
LOL. EDIT: I failed at uke shopping back when I wrote this post. Guess I really wasn't in the market. But after visiting all the uke stores on the island, by far, my favorite has to be Easy Music Center. Amazing service, knowledgable staff, no pressure to buy, a nice selection, and good prices. Hate to plug them so much, but I've just been very satisfied with the service I have received there, and the purchases I have made from them.