There is no one word equivalent in English, however it is not uncommon
to say 'first thing' or 'first up' or even simply 'first'. The most
accurate is, as you know,
first thing in the morning.
1) "I'll send that information to you
first thing tomorrow"
2) "What are you doing on Thursday?" "Well,
first (up), I'm going to study some Greek."
Cheers,
Mike