How to find out if my webspace is ready for 3.2?

No – you have to check the requirements yourself. See Hosting_WordPress

You can install the Health Check plugin to check to see if your hosting environment meets the requirements.

See The Blog post about the new Requirements

Thanks! Update worked without complications. I chose the option to switch back to 3.1.4 in case it won’t work, but with 3.2 it runs better than before. Again, thanks for help on this!