I get the feeling that this has nothing to do with the classic song from the Eagles. Mostly because the song won't be released for another three years.
While out on patrol, Spider-Man encounters a strange man dressed like a cowboy. To make it even more unusual, he's riding a flying horse. [With the helicopter blades coming out a pole from the saddle he looks like he took riding the merry-go-round a little too seriously] A befuddled Spider-Man can't help but follow him. The space cowboy makes his way to the nearest bank and robs it during a money transfer. Spouting Old West cliches, he escapes Spider-Man and the guards.
Later after Peter returns home. He and Aunt May are watching the news when they learn that Desperado is now robbing the Barret Jewelry Company. Peter excuses himself, changes to Spider-Man, and quickly finds Desperado. Unfortunately he is once again unable to stop him.
Spider-Man decides to create a false news story that involves a display of Old West artifacts at the museum and sends it to Jameson to help trap Desperado. Jameson thinks they're working together, but willing to do it in the hopes that Spider-Man will be captured as well. He plans to have extra police there as backup.
When Desperado makes his play, they have arranged it so that Desperado doesn't have an easy method of escape (his mechanical horse). This proves to be his downfall and is captured by Spider-Man and turned over to the police.
The concept of a modern-day cowboy fighting Spider-Man may seem an odd choice. Keep in mind he has fought against evil magicians, destructive plants, and a giant snowman. This is light years ahead. And quick.
1.5 webs. A bit forced at the end but overall it kept Spidey in New York and was mercifully brief.