Is it possible to solder a capacitor to SMD board using regular iron?
Yes it is quite easy. If you have no experience in this sort of thing, either get someone else to do it or practice on boards that are trash. 

With that kind of part you can suck most of the solder off each pad with some fresh solder wick and then melt one lead and gently tilt the part away from th pad, freeing that end, then melt the other side and it should come right off. 

Clean the pads up with solder wick (so the part will sit down flat), solder one lead then the other to center the part on the pads. 



Strongly suggest getting a proper SMT part. If you must hack something in, there are techniques to do it safely, but I doubt that is desirable. 

Try about 320°C for the iron tip temperature. Obviously you want to use a nice thin tapered tip that is small enough that it is suited to the job and avoid melting that wire you mentioned. (both photo credits to Digikey)