And now I'm stuck - should I (and how?) get to Mukachevo from there, or how should I attempt to get to my destination?
Mukachevo? Are you sure you need that? Since there is an everyday direct train Uzghorod-Chop-Odesa, it departs Chop 19:06 and arrives to Odesa next afternoon (13:33)*. At the moment ticket for 'platskarnyi' (6 berth open compartment) carriage cost roughy 125 UAH, while 'kupe' (4 bed closed compartment) is 230 UAH. And such tickets can be bought online.
Also, in case you don't like departure time from Chop or arrival time to Odesa, you can easily reach Odesa with change in Lviv (this way, for instance, you can be in Odesa in the morning at 8am instead)
* at the end of May there will be timetable change in Ukraine, but doubt there will be any major changes with the above Uzghorod-Odesa train.
I would like to get from Hungary to Odessa by train. I saw that I could get to Zahony (near the border) easily, and then to Chop which is real close on the other side.
Yep, Budapest to Chop is easy and relatively cheap, since direct international ticket cost only 17 EUR (one-way) or double that for the return one. But, judging from your current ip and assuming you'll start from that destination (and not from Hungary) and return there the same way, we might have even cheaper suggestion for you, let us know in case you interested
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