Focus on the ball - not your shots. I'm sure you've played Tennis for quite some time (it sounds like it), so when you know how to hit the ball. When people have their emotions get in the way, then they end up losing.
Also, someone already mentioned that you should study your opponent. Here are some good ways how:
1. Hit directly to him, see if he switches to try and improve each, or if he just keeps going forehand for comfortable shots.
2. Hit some short balls in the middle (not beginning) of the warm up and see if he comes up to the net. If he doesn't, chances are he doesn't like the net - but this isn't always the case.
3. Test his backhand with a slice. If he's a two-handed backhander, chances are he will do three things:
a) Not get the slice over the net.
b) Be in an awkward, unbalanced position to return the ball.
c) Slice back because he knows of the two above.
For a one-handed backhander, they will do two things:
a) Slice the ball back.
b) Be very careful with his one-handed backhand ending in a poor return.
The advantages to #3 are these: If they hit a slice, then it'll come floating back at you allowing you to set up again without pressure of them attacking you aggressively - especially if they dislike the net.
Either that or they hit a poor shot you can attack, or it doesn't go over the net at all.
As for you, try to hide your weakness. Try using your backhand a little bit, go up to the net to practice your net game (and if you don't have one he'll think you do).
Then in match play, just focus on the ball and how or where you're going to hit it rather than the point even if you're down 5-0 on match point or something.