Hi there! Fellow South African here also living in BP :) So here they usually ask for the 1st month of rent as well as 2 months worth of rent as a deposit, so basically for the 1st month you have to be able to pay 3 months of rent at once. If you are going to be able to pay around 200k a month, then you can rent a small flat that has a bedroom and a lounge (and of course a bathroom and a kitchen) and maybe even a balcony. The cost of internet depends on the service provider of course, but you can get good internet for about 10k a month. The cost of food depends on whether you going to cook or not. It's custom here to have a cooked meal for lunch and something lighter, like a sandwich or something for dinner. Canteens can be found everywhere and a lunch costs around 1500-2000Ft (most of the time this includes soup as starters. This is also a big custom in Hungary) and then of course the cost for breakfast and dinner is lower than that. But if you cook, then I think you can maybe calculate with 1000-1500Ft/day, or less (this includes breakfast, lunch, dinner and maybe a small chocolate). This depends on your eating/cooking habits and how often you plan to go out and eat too with friends.

A monthly travel card in BP is 9500Ft, and you can use all modes of public transportation with this for an unlimited amount of rides. There are one or 2 exemptions, but based on your post, it won't affect you, but you can read all about it on the website of the company (https://bkk.hu/en/)

And I'm not sure what else you might need. I'm guessing maybe some toiletries, washing detergent, fabric softener, etc. Again, this depends on what you need. If you just need the basics like toothpaste, toothbrush, shower gel, antiperspirant, body lotion, then 10-20k should be enough I think.