The Czech sentence is in the present tense, so the English sentence has to be in the present tense as well. The word "už - already" would possibly qualify for English present perfect tense, but the Czech sentence is focused on the future saying that we start this upcoming Monday, while English present perfect focuses from the present moment into the past. Any English sentence in the past tense is also not an option.
I have no idea if I answered what you were asking for, if not, please ask again. :-)
I got this right on the second try with "Are we already starting on Tuesday?" While that sounds weird in English, I think it sounds marginally better than with "already" at the end. But as lots of others have (already...) said, "already" just doesn't work in this sentence on the English side.