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The official price of Ukrainian international passport is in the range of about 50-100 USD because the time required to get the passport varies (from several days to several weeks).
Also, our usual advice is not to send any help to person in Ukraine before you are absolutely sure that the person is real (you meet the person face to face).
All the costs are not so high that can stop someone from meeting you especially if you help afterwards. Otherwise you are doing it on your risk.
You can fly to Donetsk international airport (there are direct London-Donetsk-London flights available). The schedule of buses running from Donetsk to Antratsyt. The distance between these two cities is only 129 km, so you can go by taxi too, it will be faster, more comfortable but more expensive.
There are not a lot of hotels in Antratsyt, so I would recommend to stay in Donetsk, and visit Antratsyt just during day time. We have some info about quite good Donetsk hotels on our page about the city. You can use the following service to translate the sites from Russian/Ukrainian if necessary.
International students can work while studying in Ukraine, there are no restrictions of any kind. Everything will depend on you, if you can both study (have good grades) and still find some time to work (may be at evening/night time, etc.), it is possible.
Of course the faster you learn the language, make friends, the higher are the chances to find a job. Also, if it is a university with a lot of foreign students, they probably can help you to find a job too, or at least give you good advice.