This does indeed sound like a hosting issue. I would ask them to raise your PHP limits.
As for upgrading manually, your absolutely welcome to simply overwrite the old theme folder with the new one. All your settings are saved in the database, not in files and should be uneffected. The only thing to worry about is if you’ve manually and directly editing your theme files. This is very involved so you would know for sure if you have.
In any case, better safe than sorry. We always recommend backing up your work first. So backup the old theme folder before uploading the new one. Also, you’ll be happy to hear that updates will be automatic from here on out so you won’t have to hassle so much: