"Ariseth" and "arose" are both translated from the word zaw-rakh'. Strong's Hebrew Dictionary defines it as:

"A primitive root; properly to irradiate (or shoot forth beams), that is, to rise (as the sun); specifically to appear (as a symptom of leprosy): - arise, rise (up), as soon as it is up."

Strong seems to indicate the word means "to irradiate" or "to appear". Strong does mention "rise (as the sun)", but the rise meaning seems to be limited to the sun or something similar, and it may very well be that it's included because this is an English definition of the Hebrew word, and in English we talk of the sun rising.

"German Luther Bible says, 'Die Sonne GEHT auf und GEHT unter und LÄUFT an ihren Ort, dass sie wieder daselbst aufGEHE.'"

This German translation says the Sun "rises", "goes under", and "runs", if I understand the words (I don't know German). All this shows is that the expression of the sun's movement is similar in German as it is in English. That's not surprising as English is a germanic language. Of course the real question is when and where did these types of terms first appear in language talking about the sun, and how and when did the usage spread through various languages and regions?

"goeth down" is from the word bo. Strong defines it as:

"A primitive root; to go or come (in a wide variety of applications): - abide, apply, attain, X be, befall, + besiege, bring (forth, in, into, to pass), call, carry, X certainly, (cause, let, thing for) to come (against, in, out, upon, to pass), depart, X doubtless again, + eat, + employ, (cause to) enter (in, into, -tering, -trance, -try), be fallen, fetch, + follow, get, give, go (down, in, to war), grant, + have, X indeed, [in-]vade, lead, lift [up], mention, pull in, put, resort, run (down), send, set, X (well) stricken [in age], X surely, take (in), way."

If I understand, Solomon describes the sun's actions as irradiating, going, and hurrying to where it irradiates. I can see why translators use the more common English phrases of rising and setting (or going down) rather than the types of phrases common in Solomon's time and locale.