Just to confirm - there are no creatures in the tank at the moment - axolotls or fish?
Once the new filter media arrives, just turn off the filter, remove those kitchen sponges then rinse the new media under the tap and put them in.
Did you get a manual with the filter? If you didn't, you can download it from here
https://support.allpondsolutions.co...articles/44001284433-700if-instruction-manual
You don't actually need the UV filter, but if you do use it the bulb must be replaced regularly. The instructions don't mention if the UV light can be turned off, or the filter run without the bulb. UV light can be used to kill the algae that turns the water green, but it does depend on how efficiently the filter moves the water past the bulb - and whether the bulb is under 6 months old.
The easiest way to get rid of the floating algae is to totally wrap the tank in something thick so that no daylight/electric room light can get in. Leave it there for 3 days. That starves the algae of light and they die. But they will come back if the cause isn't rectified.
Once that is sorted you can think about cycling the tank. This can be done in 2 ways - fishless cycling or plant cycling.
I know that axolotls need cooler water than tropical fish (maximum 20 deg C/68 deg F) so if you decided to do a plant cycle, you would need plants that can live at this temperature. Members who have cold water tanks should be able to suggest suitable plants. As axolotls tend to uproot plnats growing in the substrate, floating plants would be a better option. Plant cycling involves planting the tank then waiting until they are showing signs of active growth.
Fish less cycling is described here
https://www.fishforums.net/threads/cycling-your-new-fresh-water-tank-read-this-first.421488/ Because the bacteria grow better in temps in the upper 20's C, I would raise the water temp during cycling, then turn it down once the cycle finished.