Learn English – Am I using ‘the’ too much in this sentence

I believe that this can answer your question.

same
^{adjective, adverb, pronoun}
Same can be used in the following ways:
as an adjective (after "the," "this," "that," "these," or "those"): We both went to the same school. ♦ Our new competitors are those same people who once asked us to help them.
as a pronoun (after "the"): I'd do the same if I had the chance.
as an adverb (after "the"): The twins always dress the same.
The same is often followed by "as" or "that": Three-twelfths is the same as a quarter. ♦ I've got the same problem as you. ♦ It's the same film that they showed last year.
In informal spoken English "the" is sometimes left out before same: We'll meet again next week, same time, same place. But in written English same is almost always used with "the," "this," "that," etc.

The part "In informal spoken English "the" is sometimes left out before same: We'll meet again next week, same time, same place. But in written English same is almost always used with "the," "this," "that," etc." confirms my intuition that many people drop the the before same in real speech.

However, in written English, it's clear that we should use the same (or this, that, etc. same).

Both Riley Francisco and HostileFork have given you different answers. In my opinion, I feel leaves on Monday sounds more correct than leaves Monday. Similarly, arrives on Sunday sounds better than arrives Sunday.

This does not mean that people will still omit the word on.

I believe people will understand the meaning you are trying to say without using the preposition. However, a preposition simply enforces the meaning you are trying to convey.