Would it be better to require this authorization when the user first lands, or let them play the app first and only prompt them once they try to enter the sweepstakes?
I'm not sure I understand your question, but this seems slightly similar to the $300 Million Dollar Button issue to me:
The designers fixed the problem simply. They took away the Register button. In its place, they put a Continue button with a simple message: "You do not need to create an account to make purchases on our site. Simply click Continue to proceed to checkout. To make your future purchases even faster, you can create an account during checkout."
The results: The number of customers purchasing went up by 45%. The extra purchases resulted in an extra $15 million the first month. For the first year, the site saw an additional $300,000,000.
Basically, don't get in the way of the user experience until/unless it is absolutely necessary.
If the app does something besides the sweepstakes, it might make sense to require authorization at the moment they want to enter sweepstakes. This way your users will first make a connection (a bond) with your website and then authorizing your app with facebook will be perceived as a less of a barrier ('I know something about the app, therefore I trust it more').
Another option is to let your users enter the sweepstakes, but ask for authorization to see the results (at the very last step). This way you might get even more users authorizing: they already did what they wanted to do (entered sweepstakes) and results are there (or a user thinks they are), so the only little thing left to see the results is authorize with Facebook (which is free and requires little effort). It is psychologically much easier to continue an action you already started (or nearly finished) even if there are slight obstacles like authorization, than to start a new action that requires going through the obstacles first.
It is also important to describe to your users what happens after facebook authorization, so they won't expect anything that is not going to happen. Don't request any permissions that are not necessary, because it might lower the amount of people who actually go through the authorization process. For example, if your app isn't going to post to the timeline, don't ask for permissions to post.