What does instant mean in this context "I shall endeavor to do this Monday instant"
That's not correct English.
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I think you can make as much sense of this definition from the Oxford dictionary as I can.
dated postpositive (in business letters) of the current month.
‘your letter of the 6th instant’It's rarely used these days. One of the first hits I found in Google was written earlier than the 15th century. You also need to specify an object for the verb 'do'.
eg "I shall endeavor to do the work / it . . . "