NO! Ukrainian priests cannot marry. However, a married man (once he IS married) can become a priest.
I knew a married priest and he told me that after Seminary they had so much time that if they wanted to get married (this one had a high-school sweet heart) they could up to a certain time before ordination.
I am also close with a particular family whose father was a Ukrainian priest.
My home church now has a priest with a family.
Here’s what I was always taught: Unlike Roman Catholics where the diocese and ‘Rome’ provided for the needs of their priests, the Eastern Rite was not so fortunate. The Holy Father did not provide a means of income. So, in the old country, when a priest was given a church, there was usually some land around the church (a farm) for that priest to support himself. And for the sake of necessity & survival he would have a family who would work the land and help support not only the family but the church as well.
Then too, in the old country, unlike in the US, a priest would travel from village to village to administer the sacraments, to have Divine Liturgy (which often times were at a person’s home or even outdoors in the fields - especially the underground church), to bury the dead, to baptize (what is why when a baby was born, the father would immediately baptize the child until the priest could come), etc.
I for one am ALL FOR MARRIED PRIESTS (that is again, a married man becomes a priest, but once he is a priest he cannot marry). Have you ever confessed to a married priest? A world of difference with one who understands our every day lives as to one who tries to understand our every day lives.