Okay so I'm having a bit of trouble with this.

Say you're in the key of E minor and you want to play a melody over an E minor chord. If you play E, F# and then end on G, F# is a "passing tone" correct? But if you didn't end the melody on G and instead played the melody like this: E, F#, G, A, G. Then F# is a passing tone and A is a neighboring tone right?

Here's what I want to know. Can I play those notes over an E minor chord in that order or should the harmony switch to G major when the melody reaches G? Or can you do both?

I tried to word the question as best as I could.
Theory helps us understand music, it is not dictate rules on how it is should be used.
So yes, you can do both