Older equipment, particularly from the "capacitor plague" era (1997--2007), tends to develop leaking electrolytic capacitors. They may also dry out without any visible symptoms.
Capacitors are fairly standardised in terms of capacitance, voltage, temperature rating and external dimensions, so just try to match the label. Note that the capacitance of electrolytics has large tolerances (typically +/- 20 percent) and they’re used in places where it should not matter, such as for filtering and bypassing. Don’t forget that too high or too wide capacitors may stand in the way of something but look for a higher temperature rating or even capacitance, if available and possible.
Try to source Japanese capacitors and avoid brands known to be crap.
A basic soldering iron is the best tool for the job. A soldering gun will also work but it’s harder to use correctly (turning it on carefully and avoiding overheating). A heat gun is almost useless here, it will just reseat nearby SMD parts askew and may possibly cause some heat damage.
Removing and reinserting is done similarly: pull or push one leg at a time while heating the solder in the respective hole, wiggling the capacitor in or out. Shorten the leads before inserting and make sure to note the correct polarity.