Where are you coming from? How long will you be in the city?

I'd normally suggest visiting Barton Springs, but Thursdays are cleaning days... It is open 5am to 9am, then officially 6pm to 10pm. If they finish cleaning / refilling earlier they will re-open as early as 4-5pm. There is no charge to visit during those times.
If it is any other day, there is no charge from 5am to 8am, and from 6pm to 10pm. You could arrive at 7:55am and stay as long as you'd like (many people do that). Otherwise normal fees apply.

If you want something cultural, definitely visit the State Capitol building; there are tours and/or you can wander around on your own. If you want to visit the dome, you'll need to contact your representative in advance.
The LBJ Library is also worth a cultural visit. It isn't cheap, but it is exceptional.

You could walk around Town Lake / Ladybird Lake. There is a nice path that goes most of the way around (and you'd be very close to Barton Springs at one point). There are a few places where you can rent kayaks and/or SUPs, if you're into getting on the water.

Whenever I'm in town, I usually end up eating at Le Cafe Crepe, and have never been disappointed. Some people might suggest Hula Hut (it would be a very healthy walk, or you could get a ride); I've been there a few times and it is fine (nice location on the lake).

If you want bats, I've had much better luck viewing them from McNiel in Round Rock; they come out from under the I-35 bridge. But it is way out in suburbia, with really nothing of major interest nearby (downtown Round Rock is about 2 miles away, which has some restaurants, and I'd highly recommend Round Rock Donuts.)