Hi adastrame,
I asked that exact same question in the Slack channel. Looks like it's not possible.
Here's how I solved the situation:

• Set up a blank Bolt 5 project with composer, using SQlite DB.
• Copied over the Bolt 3 contenttypes definitions and Bolt 3 theme, then fixed all the errors that popped up
• Re-created all contents from Bolt 3 (server) in my local Bolt 5 copy
• Deleted the whole server (inkl. Bolt 3 mysql DB)
• Uploaded everything from my local Bolt 5 via FTP.

I know it's not pretty but I could not find another way (it's not my webspace).
Especially, it does not seem to be possible to convert the old Bold 3 DB to new Bolt 5 schema directly on the server. Looks like it's not even possible to set up a blank mysql db on the server without SSH access.

Hope that helps!
Best,
Felix