First of all does this look exactly the same as yours ? If not the job is still possible, just harder.
Either way, you use the pin punch to punch out the pin in the slot in the second picture.
Now getting it out. No you don't need the old tumblers. Once you get the pin out of the end the cylinder will be free. I used a pick. Push the pick in the key hole and just try to move the first tumbler up an down until the cylinder will slide out of the hole, then move to the next etc. It will come out. Plus you can see in the first picture a rectangular opening where you can push the tumblers down with a small screwdriver. The aim is to line the tumblers up so that they cylinder can slide out. If the back of the cylinder isn't the same then you will have to put your tumblers into the old cylinder. Just make sure that you track exactly where you took them out and put them back into the same place in the new cylinder