Do I need make reservation for Galleria Borghese and Castel Sant’Angelo?

Borghese Gallery always requires prebooked entry
Closed on Monday

https://www.tosc.it/artist/galleria-borghese/galleria-borghese-2253937/

Castel Sant’ Angelo
no
Not really ever crowded

Yes, you will need to pre-book a timed entry for The Borghese Gallery.

For Galleria Borghese, definitely yes. A pre-booking is mandatory for that museum regardless of season.

https://www.tosc.it/en/artist/galleria-borghese/

I wouldn't think it necessary for the Castel but maybe someone else will weigh in on that.
http://castelsantangelo.beniculturali.it
https://www.tosc.it/en/artist/museo-nazionale-castel-sant-angelo/

Editing to add: someone did (weigh in on that) while I was still copying links. :O)

Galleria Borghese - YES, you need to purchase advance timed tickets. Purchase them ahead of time as soon as possible, They sell out quickly. If you can't get them through the official website, you can get them from third party vendors. We used Get Your Guide. You will pay more than the official website, but if they have tickets you will get them.
Castel Sant'Angelo. NO. we walked right up, waited a few minutes in the ticket line and bought our tickets.