The taxis in Budapest used to be notorious for ripping off tourists. These days taxis have become more reliable and you do not have to be afraid to get into one of them. One of the confusing things is that taxi companies have different fares. Then the same taxi company has different fares for different passangers. If you stop a taxi on the street, that's the most expensive fare. If you phone them, you get about 15% off. If you have a contract with them, you get another discount.
Since you are not around enough to sign a contract with a Budapest taxi company, the cheapest way for you would be to book a taxi over the phone. But none of the taxi dispatchers speak any foreign language and quite often they have no tolerance towards those who do not speak perfect Hungarian. So a simple task of calling a cab becomes an insurmountable difficulty because of not speaking Hungarian.
But taking a taxi in Budapest can also involve other considerations. For example, some drivers drive like maniacs. A taxi might take you from the airport to downtown Budapest in 15 minutes. Normally, it is at least twice as much, with traffic, three times. I had friends coming in like this and they were holding on for dear life. But after all, a taxi is a means of getting somewhere fast, and that's exactly what happened.