If you're talking about a particular train or journey, you can say either "the train leaves at 1:00 am" or "the train will leave at 1:00 am." "I'm leaving town tonight. My train leaves [or "will leave"] at 1:00 am." It's more common to use present simple, but both are okay.
If you're talking about a regularly scheduled train, you need to use present simple. "Every day, the train leaves at 1:00 am."
(Note that you don't need to say "in the morning," since you already said "1:00 am." You could say "1:00 in the morning" instead of "1:00 am," but you don't need both.)