Ok sorry maybe I'm missing something here but......if he (she?) puts the betta in a 1 gal bowl or vase, isn't that going to need cycling too? Or at least regular water changes? And what about a filter?
What I would do if I were you is:
1) Do daily large (50%+) water changes until you can return the fish on Monday (I would return the algae eater too since we don't know what type it is and if it truly only eats algae it could potentially starve in there anyway)
2) Since you said you can't return the Betta I would cycle the 20 gal with the Betta in it. This is a fish-in cycle. But since the biload is small (the waste the betta puts out) and the 20 gal has a lot of water in it, you won't have to do as many water changes as you are now. It's IMPERATIVE that you get the liquid test kit though.
3) With the betta in the tank, test the water every day with your new test kit.
Any time ammonia or/and nitrite reads more than .25 on the test, do a water change to get those levels down as close to 0 as you can. Same with nitrate over 20. Fish-in cycles can take weeks, so be patient. You are building a good colony of bacteria that your tank will need to stay healthy but it takes time. Start a cycle log on here if you want and we can help you through it.
4) During this time DO NOT BUY ANY MORE FISH
You can certainly research fish and ask on here what else you can put in the tank after the tank cycles with the Betta, but not until then.
Also, not to burst your bubble but your 10 gal tank is too small for your goldfish. General rule of thumb is they need 20 gals for the first goldfish, and 10 gals for each additional. Not sure how many you have, but if you have 2 goldfish, you'll need a 30 gal tank, etc. Focus on getting the Betta tank cycled first, but try to save up to give the goldies a bigger tank.