First off, the 24 hour "Cycle" is not a cycle at all and does absolutely nothing for your tank or for your fish except tells you that the filter is working properly...
Second, alkalinity is pretty much null and void for Betta's, they are extremely adaptable creatures so if you acclimate him well to the water chemistry (not just temp) then he will be perfectly fine. My sorority tank is up at 7.6 pH and all my girls are just fine.
Betta's tend to prefer softer, more acidic waters but they tolerate pretty much anything. You may get some fin curling with some harder water but it's not going to hurt them, just doesn't look great to us is all.
So what you need to do in acclimate him to the new water type. You have him in a cup so that's good, empty out half of the water and put 1-2 Tablespoon of water from the tank into his cup every 10 minutes or so until the cup is filled. This make take a while so you can busy yourself with something else while you do this.
After his cup is full, should take about an hour-2 hours then you can safely let him into the tank gently from the cup.
Now what you need to do is monitor your Ammonia levels, this is important in smaller tanks. Depending on the tank size, it can hold the Nitrogen Cycle just fine but because you'll have a fish in there, you don't have the time to do a full real Cycle to the tank. So each day test your ammonia level with a liquid testing kit (get one if you don't have a liquid one. Strips are no good and are not accurate) and if it's above .25ppm then you need to do a 25-50% water change, wait and hour and test again. If the ammonia is still above .25ppm, do another change and repeat that till it's below .25ppm.
Once a month goes by or about that, your tank should now have a colony of Beneficial Bacteria in the filter or BB, those are what you need to keep your fish safe. They are what turn ammonia into nitrite, nitrite into nitrate and then you take out nitrates with water changes.
If the tank is under 5 gallons, you'll have to do twice weekly water changes on it preferably 50%-80% twice if it's 3-5 gallons. Smaller tanks need at least one 100% on it in the week. So this is why it's easier to have bigger tanks, you won't need as many changes but every tank needs at least one water change a week, no matter how old it is/how long it's been set up.
And to answer your other question, you don't need to change the alkalinity, just leave it as it is. Better for your Betta to have a stable water chemistry than something that fluctuates.