Spidey would have an easier time trying to beat Superman without the Captain Universe powers than he would WITH those powers.
Let me explain. With all "hero vs. hero" questions and threads, you have to take ALL of the character's attributes into account, not just their physical abilities and superpowers. Niether Superman nor Spider-Man would want to kill the other, thus they would (as they usually do) refrain from using their full power. Of course, they are both smart enough to know to hold less back when fighting a powerful opponent.
Superman is without a doubt the more powerful of the two, and thus Spider-Man would hold little back (if he does at all). Superman, however, would hold back just enough to put him on par (or slightly above par) with Spider-Man. This will eventually result in a victory for Superman, but Spider-Man would likely not be gravely injured or anything of that sort. (Although one should note that Spider-Man has in the past defeated one-on-one opponents far above his power level, though these are often due to exploiting that foes weakness, generally out-thinking them, or plain old dumb luck.)
Now if Spider-Man was the more powerful of the two, he would be the one holding back while Superman would be using his full potential. This can spell bad news for Spidey, as Superman had a better history of thrashing opponents much more powerful than he is than Spider-Man has. Spider-Man has a highly likely chance of coming out of this fight much worse off than he would fighting Superman without the Unipower.
Either way, one would hope that one or both of the heroes would ask why they were fighting and resolve that misunderstanding with words before either has the chance of getting hurt.