Nine years won't wear out a rim. That raised my BS flag. Rim wear, sure, but age is just a number. I'd check with another shop, mechanic, or a rider whom you trust before I swapped the rim.
Did he show you how the chainring was worn? If you've changed the chains before they get too worn, you may have replaced a cassette or two, and one or more chainring(s) might be worn. If it ain't worn, you don't need to change anything. With the (new/not-stretched) chain on the small or middle ring, try to pull the chain forward and off the ring. If you can see daylight between the chain and the ring, the ring will need to be replaced. If the chain slips on the ring when you put a new chain on, the ring needs to be replaced. Those are the only two diagnostics I know of -- although someone may pipe in with another. If neither of those says "new ring," well, you probably don't need a new ring.