That depends on whether the software allows it; Mastodon uses the AUTHORIZED_FETCH variable for this. If it's active, blocked instances can no longer fetch posts via AP.
I didn't give it nearly as much thought before I started.
In terms of price, you can set a very low price for a single-user instance. Hetzner and Netcup, for example, offer very affordable VPS (these are providers I know of, but there are certainly others).
I don't see open registration as a problem either. I've had it for years, and when I get waves of spam, which is very rare, I switch to moderated mode, otherwise it remains open.
I would host software according to your preferences. I should have kept Friendica running and turned off Mastodon, but that ship has sailed for me.
For a single-user instant, it might be interesting to connect to a relay; it is important that it is moderated. Fedimins offers such a service.
I utilize the GECOS field within my Ansible scripts to facilitate the identification of these entities as being managed by Ansible.