Hi there!
I've bought this lovely plant although I don't know what it is or how to care for it!
It's recently started getting these small white spots on the leaves and I think it might be dying. I was hoping someone can help me out? The second image has the white spots on the top leaf and the bottom leave has almost puncture holes in it? Thanks!

It looks to be a Sympathetically. Also known commonly as peace lily.
Peace Lilies (Spathiphyllum)

They like partial shade. Moist but not soggy soil, along with fast draining soil. They general are really hard to kill, so no need to worry too much about killing it.

They also have pretty white flowers.

I'm no expert, but the white stuff looks like a little residue and minor dust build up. Plant leaves should be wiped down every now and again so your leaves don't experience any blockage from the sun.

I've had mine nearly 50 years.. never had any pest problems.. If you suspect it may be one, get a hand lens or magnifying glass to take a closer look.. Could possible be mealy bug nymph..
Years ago they were sold as Mauna Loa Plants. Some marketing genius came up the the name Peace Lily, and that's what they're called now by most folks.

It could also be spider mites. Just look on the undersides of the leaves to make sure they are healthy. I've also seen it where a Spath has bloomed and the pollen is white and falls on the leaves. Just sayin'.