I met them on ____ morning last month.
One would use "Monday" or "the Monday" if only one Monday existed in the previous month. And since there are usually four Monday's per month, that would not make sense. However, if it is the beginning of the current month and last Monday was actually las month, you should say "Monday moring last week".
i met them on a Monday morning last month. which means u met them once .
'the Monday' could be referring to a mutually recognised Monday morning last month.