I have been playing Limit Hold 'Em now for only a month
I'm playing 3-6 limit
I guess this is somewhat different in live games but this is quite high stakes for a rookie.
always starting with $120 and ending if I've doubled up or lost that money
Not good, at all. Play longer if you are hot, no reason to stop when you are winning. I dont also like stopping just because you lost money. if you were also playing bad instead of just running bad this would be acceptable.

should I feel okay about being up just $145
You should be proud that you are a winner although you havent played nearly as much to say anything if you are a winner or loser in the game.