I never had mates for my bettas so not sure how big the tank should be for shrimp and betta. But just want to give my recommendations for one betta. I have 2.5 gall tanks and 5 gall tanks. I do weekly water changes. I am personally not cycling any of my tanks. For me, cycling is pain. If you don't cycle then you don't really need a filter. And if you have filter you need to make sure it not too strong for your betta .
I am going to post for you a few links. One link is about water changes for different tank size, another is if you ever decide to cycle.
Also when you buy artificial plants make sure you buy silk ones just like you wrote. I don't recommend to buy plastic , because they can be sharp and betta can damage his fins.
If you ever buy live plants always quarantine it for a few weeks in the water with NO water conditioner. Plants can carry disease.
If you ever introduce shrimp to his tank make sure you quarantine it for a few weeks to make sure he don't carry disease. Even with betta i always quarantine new fish. So you don't end up disinfecting tank and plants , and gravel.
Even with betta. Be ready some time or a lot of the time there is a chance to get sick betta. So when you go to the store i guess try to pich the healthy one. Make sure it active and not laying on the bottom, swimming one one side, bloated, only visible symptoms on the body, cloudy eye, open gill, sores on the body. Bettas are pretty active fish even in those little cups.
I also really recommending to buy Aquarium salt in the pet store. And Epsom salt at any pharmacy. Both have antibacterial/fungal effect and can be used as fires aid.
Water conditioner- Prime by Seachem is really one of the best. Just in case you have ph problem, or ammonia in your water i am really recommending Prime as water conditioner. Also i would recommend stress coat. I am personally don't use stress coat. I don't have any problem with my water. But stress coat also can protect fish from infections and help when betta is sick. A lot of people on the forum recommending it.
Also always make sure when you introduce betta to the new water acclimate him . New water should be approximately the same temperature as him. If it 2-4* different if fine. You will just slowly add new water to his water in his cup let him sit in it for a few minutes then do it again for 2-3 times - this way he will get used to the new temperature/chemistry.
So you need a thermometer. I am really recommending the ones that you can stick outside fo the tank. I love them they show temp difference really fast as you changing the water .
I also really recommending to check the heater before you put betta in. Make sure it not overheating and sometimes i read people have problems with heaters leaching something in the water. So i would definitely check if overnight to see if everything is fine. And again be careful when you put new betta make sure you don't shock him with drastic temp difference.