How do I change where the beat starts in the middle of a project?

I'm trying to write a movie score, and sometimes I put samples in the project completely freeform because I'm just doing atmosphere types of sounds, and then a scene will start and I need the music to start on the downbeat immediately. Is there a way to set a downbeat in a project regardless of what the current beat is, halfway through the project?

Global tracks should be a first stop, especially related to:

• Markers. (definitely essential to ease your woekflow)
  
• Tempo. (to adjust the tempo of the played music, folley, etc...)
  
• Signature. (more useful with the type music beat)
  
• Movie. (which you likely already aware of...)

SMPTE (and SMPTE-locking events) will also be useful to be aware of.

Is there a way to set a downbeat in a project regardless of what the current beat is, halfway through the project?

No. You'd have to somehow change the tempo of a beat or two or a bar or some section in order to manipulate the grid so that a beat ends up at the precise location you want. The easiest way to achieve that is using beat mapping (just map a beat to the SMPTE location of your sample). 

If by downbeat you mean the very first note of the song (or music snippet) to play, then the beginning of a region is where the music it contains will start playing. 

You are free to position that region to start wherever you wish in the timeline.