Razor and the xenomorph start prowling the island, unless the xenomorph lacks the mental ability to know about the fight to the death, in which case it maybe hides and waits for prey, in which case it doesn't travel far enough to leave or find a trail. So that would leave Razor searching the whole island, which could take months. In which time the alien would get hungry and adapt to hunt bird eggs or something, and wouldn't pursue Razor unless it ran low easier food.
So lets say that they are both hunting each other. Once again, the distances involved make the direct approach impractical. They are looking for the other's trail. The xenomorph, umm, doesn't seem to have a noted sense of smell, and electrosense isn't mentioned, nor are diamond tails, in fact the original tail sounds more like a club, umm, is there a better source of information than the link on the first page? So the xenomorph and Razor are hunting each other. Xenomorphs leave massive trails, in multiple different environments. They don't really seem to have the knack for subtlety. Razor is probably not particularly skilled at tracking in wilderness or avoiding the same but the xenomorph seems to be a purely ambush and horde-based hunter, so really doesn't need, nor appear to possess any affinity for tracking, I think that the mucous trails actually mean that Razor will be ambushing the xenomorph...
Razor follows the xenomorph's trail, learns its habits enough to place an ambush, shoots it, it dies.
Assuming the xenomorph possesses a keen sense of smell using its non-existent nose and tongue: It hunts down Razor, waits for Razor to sleep, triggers an alarm, probably has about a 60% chance of killing a startled Razor(nevermind, aliens are scared of dreams, as evidenced by never attacking anyone while they are sleeping, except in pods which prevent dreams. Not to mention that those were queen, royal guard, and facehuggers, all known for extremes of both bravery and suicidal tendencies...) between not getting shot, not causing a shield-based repulsion effect and then getting shot, and not shot in melee with an incapacitating weapon.
Assuming the xenomorph sits in place. It makes no trail, follows no trail, its effective area is about .001% of the island. Razor does not find it, establishes a camp, fortifies it, and establishes a long-term food supply. Stale-mate or the xenomorph gets hungry and eventually wanders into Razor's home-territory. Familiarity gives Razor the edge in spotting it, it gets shot, and dies.
Razor gets extremely unlucky and happens to walk directly underneath the xenomorph, gets jumped on. Xenomorph makes a loud noise and splashes drool everywhere. Razor raises shield, 30% chance of being too slow. Razor pushes away, 20% chance of being grabbed, 30% chance of being disabled by large object falling(some overlap). Razor shoots at alien, 10% chance of acid splash.
*notes
Razor has near unlimited raw materials, is a super-human mechanic, has a custom-built workshop, and 3 days to go to a modest-sized island where someone is trying to kill him. it seems reasonable that he will construct some means of securing a camp-site.
Newborn xenomorphs, such as the one from the first movie, do not spit acid. If they did, they would be too slow a projectile to be effective at range. At close range acid is as effective as second mouth, and therefore redundant. Acid is only effective due to its ability to destroy equipment, but the shield will take at least 5 seconds to melt, by which time the battle will be over. If the acid causes damage due to and exploding xenomorph, Razor will not be defenceless, and the xenomorph will already be dead, so it is not a factor. Acid blood is a complication that can be safely ignored.