Yes, you can!
Find an MP3 containing the sound you want to use as a ring/notification tone, then transfer it to your phone (or better yet, download it using the phone's browser).
If you download it using the phone's browser, it'll be saved into a folder called Download. Open your phone's file browser (on Samsung, it's called My Files, though it may be different on other phones - its name should be obvious enough, anyways). Find the Download folder, then select the MP3 file (usually just a long-press on the file will do it, though sometimes it's buried in a menu option depending on which phone you're using). Then, move the file to the appropriate folder - for ringtones, it's called Ringtones, and for notifications, it's Notifications, and for alarms... I'm sure you get the picture by now
These folders live on the top level of your phone's internal storage (at the same level you found the Download folder).