While applying the second coat of paint to a bedroom in my house, I ran out of paint. When applying the new can of paint,I noticed it was a shade darker. Should I prime the spots that are darker before applying the correct color? (there are several areas that need reapplying and they are noticeable)

 

You purchased paint (that they had to mix, not off the shelf) from the store, and ran out of paint say 1/2 way through the second coat, and bought some more and tried to finish the second coat....and it is a shade darker than the other half that has two coats?

If that's the case primer won't help
You'll have to paint a third coat on the "first" 1/2 with the "new" (darker) color

Just to let you (and any other DIYers reading this) know, the best way to avoid this is to buy all the paint at the same time and "case" it, or blend the different cans
This a a standard operating procedure for painters when purchasing more than one gallon of a color

If you are coming up short while you are painting (which you really want to avoid with custom colors), buy the new can before you need it, while there's still alot left of the old stuff
Then mix the old and new together to blend it

 

Since custom tints are rarely more than a tad different you may be able to paint to just paint 1 or 2 walls [complete corner to corner] and the difference not be noticed. Also note that paint always dries slightly darker than when wet. A few brands will dry lighter.