"5 months old" suggests August, so you won't be able to go to 8.13. Your licence is valid only for versions released before its end date, so probably 8.12.0 or below (but not 8.12.1 or above). Check the exact date against the release dates on https://www.atlassian.com/software/jira/update.
To go higher, you will need to renew the licence or buy a new one (renewing the ended one will be cheaper).
So, on to the upgrade. It's easier to answer the specific points
>I read that a safe way to do it is to make a backup to the database and the use a test environment where I will need to install the version from which I want to update(5.1.3) and try to update it to 8.13 and then try to scale it with my database.
You've done good research there.
You will need to go through several versions though - I'd recommend 5.1 -> 6.0 -> 6.4 -> 7.0 -> 7.13 -> 8.target. You might be able to skip some of those if testing goes well, but under no circumstances skip the 6.something to 7.0 step, you will lose data.
>To make the update I need to buy a license for 5.1.3 and 8.13 or just for the 8.13?
No, a licence covers you for all versions up to the date it expires, as above. You've got a production system on 5.1, it will become a production system on 8.x - that's one licence needed, as it's one system.
On the second question of moving to Cloud
>Update the server version to 8.13 and then switch to the cloud
Yes
> or I can switch directly from 5.1.3.
Nope. The minimum version the migration tools can handle is 7.6, and the higher you go, the better supported they are, so I'd recommend the latest version of 8 you can get to.
>My database will migrate at the same time with my software?
You don't care, you won't see it again. The important bit of a migration is that your projects, issues, users, workflows etc all migrate over to your Cloud system.
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