Just wondering how you would pronounce the following two words. I would say
Iarthar [ i-har]
Oirthear [o-her]

Is this correct?

Foclóir póca gives the pronunciations as:

Iarthar /iərhər/
Oirthear /orhər/

So mostly you are on the right track but you'll have to try to put the r's back into your pronunciation. Think of Scottish English r's and consciously trill them in Irish as well. The h's don't have to be strong but you can't leave out the r's.

Is the /r/ in these words an approximant like in English, or a flap as in Spanish 'pero' ('but')?

Hi, Riain.

According to the dictionary phonetics, they are all broad /r/s, therefore taps. I think this is because of the /h/ sound following them.

In Connemara, where the /h/ is omitted, you would get:

iarthar /i:ərər/
oirthear /er'ər/

The first /r/ in oirthear being medial slender is like a light French "j" /zh/ (I can't get the IPA symbol up here but it is the z with a tail) only up near and lightly touching the alveolar ridge, so it is almost fricative and possibly an approximant depending on your definition but still nothing like English /R/ which is pronounced way up high near the roof of the mouth with the tongue held away from the palate altogether.

Does that make any sense to you?

It does! We talked about this kind of stuff in LING 200 last year.

Now the trouble is getting the pronunciation right.

iarthar = ear-her

oirthear = leave out the C in cuir + her

Simplest way I can explain it anyway!