The Austin Library has events called 'Books and Babies' that might be fun for the kiddo, especially given the heat.
There are usually lots of other kids / babies and they have some time to just crawl around and interact after the story time is over. The schedule varies but they seem to occur pretty regularly. While there, check out a few books to take home to read to the kid. The Thinkery is nice but there is really only one area for a 1 year old to play in but it is decent sized. The Alamo Drafthouse has baby day on Tuesday before 2pm at certain theaters if the parents don't mind the whole 'screen time' thing. The lights are on, the sound is down. And it is ok to cry (you and the baby). There is a store called the 'Toybrary' where you can get a membership and check out toys. You of course don't need that, but for a small fee you can just hang out for a bit while the kiddo plays with different toys as they have areas set up. If it is not sweltering out, you can also hit up the Botanical Garden after riding the train. Have fun!