For the tank cycling, i would do a 30% water change every day or two but do not clean out the filter because it needs to build up as much good bacteria as posible to cope with the betta(when you clean out the the tank don't leave it turned off for too long), like bettamomma said i would avoid moving him around too much as well and treat him for ich/whitespot straight away.
Treating ich/whitespot can take quite awhile but don't be discouraged, i would use the med mentioned or anti whitespot/ich med by Interpet; remember to keep an eye on the levels of med in the water as the water changes will lower it so you need to top it up but be very careful not to overdose on med as whitespot/ich med is very strong stuff.
For the constipation i would avoid or significantly lower the levels of high protein foods like bloodworms/tubifex/daphinia and try feed him more vegetable based foods like cooked shelled peas and algae wafers/tablets as these will help get is digestive system working normally again.
The only other advice i can give on treating constipation is not to overfeed and in fact, feed him less than what you normally would. Remember to remove any uneaten on the bottom of the tank after 10mins to avoid him over-eating or the food rotting and causing ammonia spikes in the tank(which are bad).
I would also get a water testing kit that tests for ammonia/nitrate/nitrites to help accuratly moniter the water quality(depending on your water stats, this will also help give you an indication of when you need to do water changes and how much).
If you havn't got a heater, buy one and try to maintain the tank temp at around 25/26 degrees; if you have a filter try turning the water flow/current down as bettas don't like strong water flows/currents in their tanks.
I owuld advise you get a filter if you don't already have one as this will help cycle your tank alot.
Buying some live/real plants will also help cycle your tank
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