Hochdeutsch: "Auf der Alm da gibt's [gibt es] keine Sünde." --- on the "Alm" (see LEO) there is no sin

"Dem fröhlichen Zecher den vollen Becher" --- for the cheerful "Zecher" (see LEO) the full "Becher" (see LEO)

Let me have a crack at this. Of course, only a best-effort translation.
1. There is no sin on the mountain pasture.
2. A full mug to the frolicking drinker.

That's okay, you described it very well in English, except that the word for the two little dots is 'umlaut.'

auf - on
der - the
Alm - alpine pasture, mountain meadow (remember the Alm-Uncle in the children's book 'Heidi'?)
do (= da) - there, in that place
gibts (= gibt es, gibt's) - there's (= there is)
koa (= keine) - no
Sünd' (= Sünde) - sin

dem - to the
fröhlichen - cheerful
Zecher - drinker, boozer, carouser
den - the
vollen - full
Becher - cup, mug, stein, tumbler

I would just say

In the mountains there's no sin
A full mug for the cheerful drinker

Some words, like 'do' and 'koa,' are in Bairisch, which is a dialect spoken in Bavaria and nearby regions like Austria. So it has a sort of casual, down-home sound; but translating it as 'There ain't no sin in the mountains' would probably be too much. (-;

*f5* Which I see holger and Helmi have already said. Never mind! (-: