I'm not sure why every one is talking about killing Wolverine, per the OP Cap just has to stop him.
As with most fights I have never understood why people tend to default to a kill as being the only winning condition.
In this case the OP says he has to stop Logan, for all we know him may be able to talk him down, but even ignoring that. He may be able to tie him up or knock him out. If a bullet to the head can do it I imagine Cap with his shield has a chance. Hell somebody stronger or considerable more skilled has a shot at making him "tap out" in the metaphorical sense.
Their are other things to consider, like Wolverine breathes so what's to stop somebody from choking him out or drowning him.