I haven't read anything with AoA Nightcrawler recently (haven't read consistent amounts UXF since 12) in awhile for that matter so I am most likely rusty but I will give this a shot. From what I remember, I'd say he has this. Cyclops knows 616 Kurt to a tee and this has been proven when he has predicted Kurt's port patterns a few times and successfully attacked him. AoA Kurt I know is plenty more vicious than 616 Kurt which is an advantage only if his port patterns are different, which I assume they are, which takes Cyclops' prediction advantage away. AoA Kurt's attacks will be a lot more likely to maim and this slims the opportunities for Cyclops to tag him. I'd say Cyclops might be able to get off a shot based on chance and his spatial awareness and ricocheting blasts which has the potential to possibly tag AoA Kurt.
If AoA Kurt's patterns are the same as 616's (whether more lethal or not with the attack) then Cyclops has a lot more likely of a chance but he has to be very careful in this fight regardless as this isn't the same Kurt he has trained with for all those years.
Mystique trained this version of Kurt and this version is fully capable of (though perhaps not fully willing?) teleporting any part of Cyclops' body off. Between these two things, my money is on Nightcrawler.