Park in either Newcastle (Donard Car Park) or Rostrevor (Kilbroney Park)
If you are not fond of hike-a-bike, then take the road section from Ott Mountain all the way round to the western side of Rocky Mountain (see the map on WalkNI at the link above), rather than going Slievenamuck/Spelga/Rocky. You can ride some bits of this section, such as up the Hen Track, and down between Tornamrock and Rocky, but you'll spend a fair bit of that period either pushing or carrying the bike.
You can hook up to the trails in Rostrevor forest too, of course, but there's plenty in the Mourne Way to keep you occupied unless you are a glutton for punishment.
Ordnance Survey 1:50,000 map 29, or the 1:25,000 leisure map, will give you the whole Mournes area.
Great information about The Mourne Way, including detailed maps, on the WalkNI website