an important point about the spelling, not only does "súil" mean "eye", it sounds rather like "sual"
The letter 's' in Irish only has an 'sh' sound when it is followed by a 'slender' vowel, that is an 'e' or an 'i'. So while "siú" sounds similar to "shoo", "sú" sounds much like an English speaker would expect it to sound.

(I forgot 'guest' has already mentioned this point)

I think "shule aroon" is okay if you are singing in English, but a lot of people sing the chorus all in Irish.

The Irish alphabet has less letters than English, but makes up for it with the accented vowels and combination spelling rules. Apart from that, it doesn't have letters that aren't in English. In many cases here we now write an 'h', the sound used to be represented by a dot over the preceding consonant. I don't think that the old style lettering was especially Irish or Celtic; it's a variety of "uncial" script.