From reading similar posts I have seen a LIGHT application of neatsfoot oil recommended. As far as pasting the fabric, I wouldn't paste the fabric if it was my only hanging strop. It is good to have one with paste but it is better to have one clean and use the pasted one as needed..... IMHO.
I have a bunch of vintage strops. Most of the old barber strops seem to have been made of Shell. A.K.A. horsehide which has little if any draw. The strop may not feel like it is doing something while in reality it is.