If you have an enchanted lyre from Fremmenik Trials (I think), use that to tele to Relekka. Or use house teleport with your portal in Relekka, then walking from there. That is the fastest way I know.
go to the place in varrock, move your house to relleka and tele there
or go to zmi on the zmi world and people are always doing tele group to moonclan island
It would take you less than 2 minutes to buy a camelot teletab, break it, run north, get a seal of passage, travel with lorak searunner and then run to the lunar altar.