Castel Gondolfo -- open on 13th January?

According to their calendar, yes, Castel Gandolfo will be open: entry at 9:00; last entry at 13:00; closure at 14:30.

https://www.museivaticani.va/content/dam/museivaticani/pdf/visita_musei/scegli_visita/ville/calendario_palazzo_papale_2025_2026.pdf

It doesn't look like advance tickets are available yet on the Vatican Museums' website but I'm sure they will be at some point. That said, I wouldn't think you'd need advance tickets.

Thanks. They don't seem to be selling tickets now, however, unless I'm doing something wrong.

As I said above, they aren't selling tickets yet for your date. Keep checking although I can't imagine that you can't arrive in January and not be able to buy tickets at the site.