Translation:Luckily, I came across a hot spring.
Nuance here makes this an unexpectedly quirky exercise. "By luck" means "I was surprised to have found this unexpected thing" whereas "Luckily" means "I was hoping to find this thing because had I not, something bad might have happened."
Am I wrong to see this difference?
Par chance... = By luck, I found this silver dollar. What a nice surprise!
Heureusement... = Luckily, I found this silver dollar. What a relief!
As noted above "to fall upon" (meaning "to unexpectedly discover") is unusual but perfectly possible. Yours is subtly different, "fallen on" seems to indicate a literal fall.
Though my French is far from A1, I have lived here for a number of years, so let me make another comment: I understand the temptation to translate the passé composé into the present perfect (je suis tombé -> I have fallen for example), but very, very often the preterite (the past tense) is what is intended, so I fell is probably better (though the pp is always a possibility).
Anyway I came in to say that the proposed correct solution seems to conflate luckily and by luck which I think are two distinct concepts. To me the best translation would be:
By luck I came across (or happened upon, or even simply, found) a hot spring