Booking a tee time at the old course is quite a challenge.  Due to demand, and the hosting of The Open, St Andrews is not accepting any advance registrations for 2022.  What they do have is a ballot system that is announced 2 days in advance.  So, if you plan to be in Scotland and hope for a tee time – sign up for the ballot.  https://www.standrews.com/play/courses/old-course/booking-the-old-course

You can also show up at the crack of dawn on the day you want to play and hope for a kind starter.

For 2023 you have two possible other options.  Watch the website for the first date they will accept your advance application for a tee time and get yours in the moment the window opens.  Which may be at 1am your time.  Or, you can go through an authorized booking agency – which is your most expensive option.  My recommendation is to try the advanced tee time and then if you get it, plan the rest of your trip around your St Andrews day.

I recommend Hidden Links as one of the best booking agencies.