I have an old backup of my site. I don't necessarily want to finalize restoring it because it might take a lot more work to rebuild the site than to work from where I have now. How can I test-restore a site so that I can see what the site was like but without actually reverting the site to that backup?
You can't: you either restore a back-up or you don't. In the process of restoring a back-up, current files and database entries are moved/edited/deleted in order to go back to the previous set of files and database entries.
The easiest thing to do (rough translation of Mohsin's answer) is to create a separate website ("staging site") where you upload the backup. This way, you still have the current live site ("production") and have the staging area running in parallel - so you can make a side-by-side comparison. You can use an external web server for this staging area, or run it on a local web server that is running on your computer ("localhost"). Here is a tutorial on how you can do that.
Please keep in mind that your database is filled with links to the domain (www.website.com) and that you have to search and replace these domains throughout the database. This is a good tool for that, although (as the site also stipulates) you have to be cautious when using it, as it can mess up your database pretty bad.
Another option that is somewhat easier to newcomers is to use the All-in-One WP Migration-plugin to make a back-up of your current (production site), then restore the old version on your web-server (overwriting the current site) and then make a back-up using that same plugin of this old version. This will still require you to switch back and forth, but as this plugin saves your files and database in one file, it is much easier to make and restore a back-up. Also, it provides a database search-and-replace option (which is a little more foolproof than the one I linked to earlier), in case you still decide on migrating any version to another domain/server or your localhost.