The fold might be a problem - so might the dryness if it is dry to the point of the surface crumbing.
You could try using saddle soap and water - get it wet and really lay into it like Mikael says with a stout, seamless/patternless bottle - pushing the bottle up and down, not rolling it.
Towel it off and then place something absorbent and without a heavy texture to it around it, and press it under weights like Mikael says. Probably be nearly dry if left overnight, but you may have to let it dry in the open, making sure that it does not curl (eg weighting the end).
The saddle soap will have got some moisture back into the strop - essential for the longevity of leather - as well as oils such as tallow. You could repeat the process to add more oil - it only adds a little, which is what you want (you can add more, but it is hard to take it away!) or rub a small amount into the back of the strop and leave it overnight again, testing the draw.
If you know you want more oil than just saddle soap will add, putting it on while the strop is still damp after saddle soaping is best - pour some (not a lot!) on a balled-up rag and work it quickly into the surface, rubbing clean witha dry rag. The leather, being damp, will not absorb a lot of the oil, so its safer to add oil to the front of the strop like this.
Bear in mind that if the strop is texture - ie has a slight nap - you can kiss that goodbye after doing the above. You could abrade it again with fine wet and dry paper when it is dry, but if the leather is that old there is a danger you will destroy it.
Regards,
Neil
