Borrowing from Russian ча́шка (čáška) [ˈt͡ɕaʂkə] (EDITED: This word is common among many Slavic languages, so it is 100% not clear what exactly language Romanian adopted this word from)
How do we know it's borrowed from Russian and not the other way around? Just asking.
Because its Slavic etymology, it is clear the word was borrowed from Russian-Ukrainian (both languages have this word) or from some South-Slavic language (Bulgarian has it also). The root is "čaša" - "bowl, cup" + diminutive suffix "-ka". The Slavic word is related to Old Slavic "česati" "to scratch, to carve" (because utensils were made by cutting and carving wood). It is not connected to similar sounding word "ceai" tea.
I think it's more likable that the word ceasca was borrowed from Serbian, since Romanians had much more contact with Serbians than with Russians or Ukrainians. A lot of their culture originates from Serbia as well.
Actually it was borrowed from Old Church Slavonic though Ukrainian (not Russian!) with Bulgarian and Serbian influenced on Romanian too.
These official Romanian dictionaries say is was exactly Russian:
Dicționarul explicativ al limbii române, ediția a II-a 1998
Dicționar universal al limbei române, ediția a VI-a 1929
Dicționarul limbii române moderne 1958
Noul dicționar explicativ al limbii române 2002
But "Dicționarul etimologic român 1958-66" shows general Slavic as a source and compares it with Bulgarian and Polish but not Russian or Ukrainian.
Probably, Ukrainian language wasn't even standardized when this word was borrowed.
Can you show a dictionary where exactly Ukrainian not Russian, was shown as a source?
It would be better to use the word "specifically" than the word "exactly" in both of these applications..