however, if you want to go for it, you will increase reliability of the circuit:
the equivalent series resistance (ESR) of a filtering capacitor will determine the voltage ripple at the output of the VR: poly has good ESR (lower than electrolytic).
you can decrease a bit the cap value (=partially compensate for space loss), but the tank charge required during voltage transients will limit the minimum poly cap value: i would not go for less than 60-70% the capacitance of the old electrolytic cap.

lower capacitance value will lead to higher output voltage noise (your screen may not be working just because the capacitance is low)
higher capacitance may result in high current spikes at start or VR instabilities

choose, as replacement, the same techology, voltage and current rating and capacitance value (no need for same brand); if possible, go for higher temp rating (105+°C), "long life" rating, lower ESR value, 10% or 20% should be fine.
reliable caps: tdk, avx, panasonic, epcos, kemet (electrolytics); epcos, kemet, wima, avx (poly).

how to unsolder: three solutions
a) cut caps leads (and throw away the component, leads will probably be too short for mounting it again), so that you just heat up (1-2s) each pad and pull out each pin with pliers while tin is hot.
b) use some soldering wick: heat up the wick and THEN place it between the tin to be removed and the soldering tip; don't heat up too much (1-2s, then wait 4-5s for cooldown); repeat until you can remove the component.
c) use a soldering sucker: heat the pad until the tin melts, then suck it :)
(postprocessing: use steel wool if you want to clean the surface after tin removal - i never did).
how to solder:
- melt a small amount (i said small) of tin on the tip of your soldering iron (so you've already heated it up)
- place the component, then heat the PCB pad and the component lead, the tin will distribute evenly on the pad surface thanks to the surface tension. be careful not to touch the FR4 or heat too much and damage other components
- if you need more tin, shove it between the pad and the lead, not on the solderer iron tip (you want to keep it hot); if the tin does not melt, temperature is too low.
note that the flux on the tin wire is active for ~1-2s after melting. also, surface will be wet only if temperature is high enough, the surface is clean and there is enought flux to prevent surface oxidation
you don't want a cold solder, as in here (source: Wikipedia):

correct amount of tin: stop before you have a bulb joint, you want a slightly convex surface (source: llamma.com):

too much, ok, not enough, not enough

cuckoo

block scheme
active input filter (PFC) app note : numeric example at the end of the document
PFC + DC/DC stage example