To pedal or not to pedal, that is the question... and there really isn't a good single answer. First, with regard to the melody (for once, clearly at the top!) it is almost always either 4 or 5. There are a number of places, however, where you do have to switch fingers quickly -- all the places where the melody is in quarter notes, for example. On the other hand, those are exactly the places where you do not want a great deal of pedal -- if you use pedal there at all, it must break with each quarter note otherwise those passages will become mushy. But the fingering is relatively easy: if it's descending, strike the quarter note with 4 on the beat, switch to 5, and carry on. Ascending just the reverse -- strike the note with 5, switch to 4 and carry on. It can be played legato, without the pedal at all, and it wouldn't be a bad idea to practice it that way at times.
However... the piece is clearly, very clearly, intended to be played with pedal throughout, at least in my humble opinion, except in one or two places where the bass has a figure in eighths or a trill. However, the pedal must be released and reapplied at each change in harmonic structure, quite faithfully, otherwise you will get mush.