There are two preconditions if you want to update a software package via
apt. First, the software's publisher needs to build and release the software as debian packages, which can be recognized by the file ending
deb. Secondly, the publisher needs to operate a repository. That's basically a url from where you can download
deb packages, plus some metadata like which version is the newest and so forth.
apt update and
apt upgrade (or
apt full-upgrade) basically walk through the list of repositories you have configured on your system. For each, they check whether there are newer versions of
deb packages you have installed, and offer to install those newer versions.
The team behind PrusaSlicer does not seem to offer their software as debian packages, and they don't seem to operate a repository. So you can't use
apt with that software. There's an issue (which is the Github term for "support ticket" or "bug report") suggesting to basically add a repository, but we'll have to wait and see when or if they're going to do that.
As far as the error message from the AppImage package is concerned, packages (debian packages, AppImage packages, ...) typically don't contain all code necessary to run the program. For example, PrusaSlicer didn't reinvent the wheel and write their own graphics code to draw windows, buttons and the like on the screen. They're using an existing library for that, in this case the graphics toolkit GTK. Obviously, GTK needs to be installed on your machine so PrusaSlicer can use it. That's what the error message is complaining about, "I try to use the library
libgtk-x11-2.0.so.0, but I can't find it". pilot6 already mentioned in his answer which debian package from the Ubuntu repositories you need to install to get the library.