You guessed it. By now an iPod 5th gen second hand would be around 3 hours of battery life in my experience.
You can go for the battery replacement. I've already replaced the battery in my ipod 3rd gen twice.
You can also even go to the Apple store and at the genius bar ask if you can pay for an out of warranty battery replacement ipod. These can cost around 50 bucks. I would say thats a great deal if the only problem is the battery. You can get a new unit (same generation) just for exchanging the old one + paying 50.