Translation:I am getting up early these days.
Éirigh is one on those Irish words with many meanings. Don't get too hung up on literal meanings. There are plenty of English idioms that would make no sense to a new learner of English, we just are used to them and do not notice. For example my use of "hung up" to mean stuck. No one is hanging anything up in that idiom.
Duolingo doesn't support alternative answers for "Type what you hear" exercises - they are really "Type what she said" exercises. (Actually, they're "Type what's in the script" exercises, because the text-to-speech engines that Duolingo was designed to use don't go off script).
Unfortunately, Táim ag an Tá mé ag can be hard to distinguish because the é and the a naturally merge.
Yes, she does. While she normally pronounces laethanta as written, she has a tendency to use aí endings for plurals, a reflection of the fact that there is a wide variety in plural endings in Irish, and sometimes different dialects have settled on different endings, and most of the time this doesn't cause any confusion (except, obviously for learners).
It's a bit unusual in this case though, because the plural of lá is already "irregular" - it's not like going from spúnóg to spúnógaí instead of spúnóga.
I'm having trouble with her pronunciation of "éirí".
Too much of a "d" sound and not enough rolled "r".
See http://www.teanglann.ie/en/fuaim/éirí for comparison. The Connacht speaker sounds much like her but with less d-ish of a sound.
(for some reason the URL that duolingo auto-links lacks the specific word, which will need to be added separately.)
There are 5 different phrases that include éirí - in the other examples, the Munster speakers don't have quite the same r sound as Sinéad uses for éirí.
In this case, the "d" sound that you refer to is just an artefact of éirí being squeezed between two g sounds. It's just part of the natural variability of speech.