You'll be fine.
Take a look through the links on this page, which includes the option to request priority processing. I will cost you more, but in your case may be worth it.
I'd even think that a normal application would make it in time. Try and get as much done as you can tomorrow, including all aspects of the application that require your presence.
From memory you will require someone who knows you to sign the back of the passport photos to verify it is you, and also sign the form - keep that in mind if you're trying to do it all tomorrow.
There is also an online application form system these days - have a crack at that tonight and then go from there.