Yes, anyone can go. It's not appointment based, although it can get busy at certain times. You can give them a ring on 0115 958 1634 and they can let you know the best times to come. I'm fairly certain they don't charge to look at the archives, but have different fees if you want copies.
They are open Tues - Sat with Tuesday's being open til late and half day Saturday.
Hope you find everything you need.
