Getting started life in Japan

Hi.
Tokyo is pretty expensive. According to the internet, a single person living cost can be around ¥121,212 without rent. 

Also you can check out this post: https://planetyze.com/en/japan/forum/japan/living-in-japan/costs-of-living-in-japan-tokyo
And this video below:



Let’s consider the renting apartment cost, let’s say you live outside the city center in Tokyo, so count that as around 80,000yen for a small single apartment. That's before the utility cost that can be around let's say 10,000¥-20,000¥. If it's winter the cost can get a little higher if you turn on the heating. 

Most apartments here are unfurnished (no fridge, no washing machine, no stove, etc) so you’d have to buy them yourself. If you want furnished ones, it’d be way more expensive. You need to buy electronics as well, wifi, and other things. 
If you want to save money, consider buying stuff from 100-yen stores (there are lots in Tokyo), cook your own food or eat in chain stores that are less expensive, buy secondhand electronics rather than new ones. 

Hospital cost can be very expensive so health insurance is important (it's required for long-term residents). 

As for jobs, yes there are jobs for foreigners here but can sometimes be difficult to find. A job that is a little easier to find is teaching English, or maybe even Hungarian (if you can find one).

But, make sure you have a work visa.

To find jobs you’d also need a SIM card which mostly here,  the common one is you buy a phone and it comes with a contracted sim card. If for SIM card alone, I recommend GTN for foreigners because the staffs speak English well and the customer service is convenient as well.
Learning a little bit of Japanese will be helpful as well  
So I think that if you are moving consider also the moving cost and living cost for at least 2 months just to be safe. Hope this helps you a little.