I was in St. Andrews in October 2003, got to the starter's hut around 6:30 AM (sunrise is close to 8 AM that time of year) and was second in the walk-on line. I was playing by 8:15 AM. Something to consider if you don't get a time via the ballot.
Also, for such a short trip you should consider staying the whole time in St. Andrews, even if you venture out to Carnoustie and/or North Berwick, both of which are well worth playing. That said, there's plenty of very good golf right there in Fife to keep you busy for five days, including Crail, Lundin Links, Kingsbarns, Elie and the New and Jubilee courses. There are many others on which opinions are mixed, such as the Castle Course, St. Andrews Bay and even Scotscraig, the subject of another recent thread.
Enjoy, but keep in mind that it's not easy to play 36 that time of year, which some people I encountered called "winter"!
P.S. - I can't believe no one has called Ivan on his statement that the New is better than the Old?