Its probably best to replace like for like - ideally the old felt has to be chopped off using a garden spade, then any rotten woodwork beneath should be attended to. The felt needs an undercoat layer, followed by a top coat of the mineralised variety. You can buy cold sticking compounds for roofing felts but they are seriously messy - a better job is to get a roofer to use molten pitch to stick the stuff down. Seriously dangerous!
howtomendit.com for free repair help, information and advice.
Try fixfault.com to fix roofing.