The cap you found is an electrolytic of unknown origin, meaning, god knows if it will last ten years or ten minutes. I would STRONGLY urge you to search ebay for "1uf polyester", which is a film cap, of 25, 50 or 63V rating. Film caps are non-polarised, and generally infinitely reliable.
The circuit traces in question are very, very tiny, and a solder sucker is NOT applicable, and you need an iron with a reasonably fine point tip, but with balls, i.e., 40-50W & controlled at 700 degrees F. Best & safest way to remove the stock, tiny SMD cap is to add a little fresh solder(rosin core, lead/tin blend, as lead free stuff is crap) to each end, then lay the iron down along the cap to melt both sides at once & gently nudge the cap sideways toward the top end of the little board. Then, trim the leads of the new film cap to length & form so as to be able to lay it more or less down & spaced to match the solder points of the old cap. Next, flow some fresh solder onto the end of each cap lead, so that the end is coated, but not blobbed. Add a bit of fresh solder to the iron tip, hold the cap over the two little board connections, then melt each wire into each spot. Finally, add a bit of silicone rubber caulk/adhesive between the cap body & board to give it support, so as not to stress the board traces by flapping around.
Above all, BE DELICATE & VERY CAREFUL.