Cheaper, definitely self-hosted, but as with everything, you get what you pay for, you'll have to take care of fail-over and backups on your own.
Start with a small instance on DO, enable backups (for a small fee DO will create weekly backups of your instance), as your project grows, tune PgSQL and resize the instance up, add a slave, weekly backups of the instance might not be enough at this point, so also pg_dump the database at a more frequent time interval (1/day), send the backup to 1 or 2 offsite stores (S3/other remote server/etc).
Managed, comes with it's own problems as well, but at least you can blame someone else when it breaks :).
Start with a small instance on DO, enable backups (for a small fee DO will create weekly backups of your instance), as your project grows, tune PgSQL and resize the instance up, add a slave, weekly backups of the instance might not be enough at this point, so also pg_dump the database at a more frequent time interval (1/day), send the backup to 1 or 2 offsite stores (S3/other remote server/etc).
Managed, comes with it's own problems as well, but at least you can blame someone else when it breaks :).