Vatican museum

Do you need to buy tickets in advance?

Yes, if you want to skip very long ticket lines.

Does the entry ticket include the Sistine Chapel and St Peters
Basilica.?

Tickets for the Vatican Museums includes the Sistine: it's IN the museums, The basilica is free so no ticket needed although you do have to stand in queue for security checks (you have to go through a security check at the Vatican Museums as well.)

How much do the tickets cost?

See the Vatican's website:

http://www.museivaticani.va/content/museivaticani/en/visita-i-musei/scegli-la-visita/visitatori-singoli.html

For an adult "open tour" ticket purchased online, you'll pay € 20.00 (€ 16.00 + 4.00 booking fee).
You will need to choose a specific day and entry time when you purchase your ticket.
"Open tour" means touring the museum yourself.

If you wish to book a guided tour, then you'll pay a higher price: see the options on that same page.

If you go at 1pm is there enough time to see everything ?

You probably could spend a month and not see "everything", but 4-5 hours should be enough to see the highlights. After that long in a museum, I suffer overload and my eyes glaze over and my brain goes numb

I found Vatican 'Happy hour' tours on Friday evenings and signed up for one of those. Thinking that there will be significantly fewer people. Think that was E35 and included what sounds like heavy appetizers