So, do you have a soldering iron? Or can borrow one?
To remove the blown white cap, you don't need a solder sucker, just the iron. Put finger pressure lifting one end of the cap upwards away from the PCB, while you melt the solder on the pin under that end. Helps if you can hold the circuit board in something as you do this, unless you have a third hand/helper. The cap will lift a little bit. Switch to the other end, do the same, it will lift a little more. Repeat, going back and forth, and the cap will come free after about 3 cycles.
You can clear the holes of solder by poking them with a toothpick while melting the solder with the iron.
Try to keep all the solder-melting time to a minimum, or the heat can make the copper track separate from the PCB. But it's not a disaster if it does, as that can be fixed with a little bit of wire.
Once you get the burned cap out, clean as much as you can of the mess off the nearby parts. Methylated spirits and a tissue will do.
Are you sure there are no other fuses? There might be one in a pop-out holder at the mains cord entry socket on the outside of the machine. If not, do check the circuit breaker for that power point didn't trip. Plug a lamp or something else into the power point to check.
Also, can you post a photo of the other side of the circuit board for the area you showed above (and a bit more around.) So we can check that cap, fuse etc are really what we thought.