Kikuyu spreads rapidly through rhizomes and stolons. Grass not only grows upwards but it spreads outwards too. Varieties like Buffalo or Kikuyu are notorious for this and they will throw runners out very quickly.
Traditionally gardeners would dig in garden edging to prevent the grass spreading. This allows you to use a line trimmer run hard up against the edging to keep the grass in check.
You might like to add some weedmatting along the ground and up the fence to block sunlight before installing the edging.
Unfortunately, if grass is coming from under the neighbouring fence, you will have a difficult time and might have to look at poisoning it. But poisoning the runners coming under the fence could kill off some of the lawn on the neighbour's side. It's best to check in with them to see if they'd like to control the grass on their side first with edging before you go ahead and poison the grass on your side. - MitchellMc
I used a 380 mm roll of Dampcourse. Seems to be working OK so far. Kikuyu roots are insidious, traveling underground down to at least 300 mm.
Once you stop the Kikuyu from re-infesting the area, you need to remove any already in the space you are protecting. This is a long and tedious process, as any pieces of root left in the ground will re-sprout and spread. - andring