Do cycle it. The opinion there is from someone who's likely of the school that cycling fish in with a low bioload is OK as long as the fish doesn't die. While it's true that a low bioload of fish can sometimes survive the cycle, they will experience discomfort from even small amounts of ammonia and nitrite in the water and potentially shorter lives/damaged immune responses as a result.
Bettas don't like a lot of water movement. This is partly because of their native habitat which is densely planted still water, and partly because the Betta splendens we see in shops is very far removed from its wild relatives. Line breeding to create those glorious long fins has made it hard for them to swim and particularly to deal with currents.
They do, like all fish kept in tanks need a filter. Without filtration ammonia will be a permanent presence in small volume of water. Those kept by some breeders in unfiltered tanks are able to survive it because they are given massive chemically and temperature matched daily water changes and kept in virtually sterile conditions where no food or waste is allowed to accumulate and the tanks are usually bare of any substrate, decor or planting. Not a nice way to live!
When you have the tank set up with filter and ready to go, switch the filter media for a piece of sponge form your goldy tank. Then dose with ammonia as though cycling and see how well it copes. You may find it is instantly cycled, or at the least, the cycling time will be shorter.
Take a bit of the goldfish filter out and swap it with the filter for the betta tank. The bacteria will gradually spread to the new media over a couple of weeks (as you've culled some of their population). Using this as a starting colony you can then put it in the new tank and start adding ammonia. You may find you get an "instant cycle" that way, but it isn't guarantee, so I'd test it out with ammonia rather than fish obviously. You'll certainly reduce your cycling time.
The sponge filter needs an air pump to work. the pump forces air though the sponge pullin water with it and circulating the water over the bacteria living on the sponge. They're very popular with betta keepers as they don't create much water movement. Shrimp keepers like them too, as the shrimp will pick morsels of food off them. (Whether you can keep shrimp and bettas together depends on the personality of the betta.)
Thanks guys,unfortunately the filter with the small tank isn't a sponge filter, had to google it but it's just a normal filter with a hollow cylinder sponge in it, which will be impossible to replicate with the sponge from my Eheim. I'm going to order a sponge filter though and go through the process, that sounds much easier and they seem quite cheap..
Any recommendations for a sponge filter?
Also what substrate would be best? Ideally I want the bottom to be covered in grass like plants so maybe thinking of peat or similar?