The only affordable way to
see Aulani is to stay on Waikiki, and drive over. You can swim in the lagoon (we did) since all beaches in Hawaii are public, and they can't boot you out. Have a lunch at the resort. See the property.
To be honest, Aulani is kind of out of the way, and it's far from everything. It's nice and all, but you will probably want to go Waikiki every night.
Here are the things we did in Oahu, and to be honest, it was my favorite over Maui, even though we stayed in a much better hotel in Maui.
1. Snorkel in Hanauma Bay. Even if you can't swim. The water is knee deep and full of tropical fish. I have never seen anything like that.

2. Climb the Diamond Head.
3. Visit the Polynesian Cultural Center. If you stay for the luau, remember they don't serve alcohol (it's run by BYU). I recommend those tiny bottles you can add to drinks inconspicuously.

4. Submarine tour off Waikiki. Where else you can go on a real [tourist] sub?
5. Watch surfers on the North Shore.

6. Hike to Ka'ena Point (that's on the same side of the island as Aulani).
7. Go to Kailua and Lanikai beach. Ah-mazing.

There is a reason why the Obamas rent in Lanikai for vacation.
8. There is a Buddist Temple on the East side of the Island. Go early in the morning when it's quiet.
9. Do a segway tour of Waikiki. Lot's of fun.
10. Explore the restaurants of Waikiki. From whole in the wall Japanese places to Duke's and Moana Surfrider.
11. The obligatory Pearl Harbor tour (that one is actually close to Aulani) and Dole plantation tour. We did not do the last.
Other than a really nice pool, there is really nothing that sets Aulani apart, imho. And you don't need to go to Hawaii to swim in a nice pool.