Yes, the 737-300's were ordered in the mid 80's not to replace the 727-200's because at that time they were still young but like you say to increase the narrowbody fleet which constisted then of 727's, 737's and DC-8's. The first 737-300 was delivered in 1987. In the late 80's, not much longer after the first 737-300 was put in service with United, an additional order was placed for 737-400's and 500's. The idea was to focus on Boeing jetliners, the 500's were delivered in the early 90's but in 1992 United looked at Airbus and said "he, the A320 would be good for us, better than the 737-400" so they decided not to buy 400's from Boeing and order directly from Airbus.
United retired it's last 727-100 in 1993, same time when American retired that aircraft, and that year received the first A320. They received their last 737 just before receiving their first A320 so they went into a "boom" in the mid 90's by increasing capacity and expanding it's network domestic and international. A bunch of 757's where phased in as well since the early 90's to replace the DC-8's and some of the 727-200's which were getting close to retirement age. So they had then a mixed fleet of 727-200's, 737-200/300/500's (yes a very large 737 fleet), A320's and 757's. The 737-200's were due anyway for retirement in the late 90's but the 727-200's were due to be phased out only this year, 9-11 is the only reason why they were phased out end of 2001, more than one year before what was expected. A319's were ordered for two good reasons, to supplement the large A320 fleet which had spare parts and cockpit commonality and to replace the aging 737-200's.
So now in the 2000's, United has a narrowbody fleet consisting or a large fleet of young A319/A320's, 737-300/500's which are a little older compared to the Airbus, and 757's. Now that the 727's and 737-200's are gone, next to go are the 737-300's but those are not so old so they should continue to be in operation for a few more years. That's of course if United ever gets out of Chapter 11. If the very worst happens, that is if United goes into Chapter 7, Continental may pick up the 500's and some of the older 300's and the A319/320's will be put for sale, some of the latter may be picked up by Northwest.
That's the whole story of the narrowbody fleet at United.