how to upgrade logic pro x 10.5

Search in the App Store for "Logic Pro".

If it shows "Open", then you are at the current version. If it shows "Update" (or so), then you can force the download of the update. If it shows "Get", then your App Store account does not have a license of Logic Pro - typically happens when an application was bought via a different Apple ID. Check that you are logged in via the "Store" menu, there you can also search for all purchased applications.

Launch the App Store, at the bottom left you will find the "Updates" tab. Select it and it should show Logic Pro.

How do I backup up Logic Pro 10.5. So I can download updates

It shows Update and, when I click on this, all I get is the "buffer" sign and nothing else happens (and I left it for ages). Internet is working perfectly.

Try the suggestions here: https://www.makeuseof.com/tag/fix-mac-app-store-problems/