The best internet wi-fi subscription in Budapest?

Just confirming something: Are you sure you do not mean broadband mobile? Unless there is a WiFi antenna nearby (100 to 200 meters or direct line of sight with a collector antenna installed on your building) WiFi services will not be a option (and paid WiFi access as your only Internet link is not what I would recommend in most cases, for a number of reasons).

If you mean broadband mobile you have two options: 3G or if you are in an area that supports it 4G. As far as I know, they all have Gigabyte tariff limits in Hungary. After you go over that limit, they slow you down. But there are extra packages you can get, such as unlimited email or web browsing that can mitigate that limit, but using Skype I do not think will come under an unlimited traffic plan.

It is odd if you are in Budapest the residence does not have a phone, as a land line over the phone line would be the best option (fastest and most reliable and secure).

DIGI.  They just increased their speed and are cheaper than UPC

I think they eventually want to access the internet through their wifi-capable devices and not much care how :-)

So you'll probably need a broadband subscription (dsl, cable, or fiber) in the apartment, and you'll connect a wifi router to that.
Type your address here to see what's available:
http://www.tantusz.nmhh.hu/tantusz/internet/keres.jhtml

Download caps in the hundreds of gigabytes a month should not be a problem unless you are constantly torrenting or netflixing, albeit I do prefer uncapped on principle. Skype video call bandwidth requirements from the horse's mouth: https://support.skype.com/en/faq/FA1417 … skype-need

I'm an IT guy with a family of 4 and we have no need for bandwidth beyond the cheapest 7Mbit ADSL there is, but I do appreciate other people subsidizing advancements in broadband technology, making the bottom tier cheaper all the time :-)

Mobile broadband: Shop around for a "trial", where you can cancel and get your money back within a week or so:
One provider might work great at your apartment, the other might be horribly slow, and there are 3. They usually offer fairly restrictive transfer caps and high overage fees though, but this might have changed since I last checked.
The advantage is you can take it with you on the go, for a vacation (within the country) or when your lease is up.