Bone skeleton Wolverine? Cap smashes his face in. Not only is Cap far superior in h2h, his bone structure also has higher durability showings as seen when he took numerous punches to the face from Bucky's roboarm without getting fractures.
Wolverine will heal any cut and bruises but he's going to get himself knocked out.
If it was adamantium Wolverine it'd be more interesting.
How though? Logan hasn't gotten KO'd since X2. And the forces he's tanked have been far more than what's actually knocked him out. Juggernaut. Silver Samurai. X-24. Sabretooth. At this point I don't think Cap can knock him out. And if he can, not fast enough to do it before Logan scores a good few cuts or stabs on him.
OP states no claws. Cap has taken hits from Bucky, Spiderman, Ultron, Ironman and even Loki and was able to spring back up to his feet. Wolverine just doesn't have the power to seriously hurt Cap without his claws, even if he had his adamantium.
Round 2 Adamantium Wolverine makes it much closer, though I'm still going Cap. Kinda wishy-washy on that though.
Even bone skeleton Wolverine has survived stuff like the Sabertooth bar/lumber tractor trailer fight, the Nagasaki A-bomb, and Magneto snaking rebar through him.
Really the question for both rounds is this: can Cap do enough damage in a short enough amount of time to overpower Logan's healing factor and enhanced durability in Round 1 and then the same + adamantium frame in Round 2?
I tend to lean towards Frothbyte in that Cap has taken and dished out plenty of hits on guys WAY WAY above his pay grade, so, again...
Round 1: Cap in a tough fight.
Round 2: Still Cap, though I could even lean towards Logan taking a few now.