Cup stains?

I’ve been noticing that my cup has stains around the rim. I’m not too sure how, as the cup doesn’t even fill up that high, and even if it did, I always rinse it every time I take it out. As much as I’d like to think it doesn’t bother me (because I’m the only one that sees/uses it), it kind of does. Is there anything I can do to lift the stains? While I’m here I’ve also noticed blood in the little holes at the top. I’ve tried to clean them by filling the cup with water, putting my hand over the top, flipping it upside down, and squeezing the water out through the holes, but it doesn’t seem to help.

UPDATE (with pics): I emailed Lena and asked them for advice, because I knew they would know what works best. I picked up some hydrogen peroxide (3%), a new toothbrush, and started scrumming it with the peroxide before and after I boiled it. I’m so so happy with how it came out, there’s still a very slight ring but I’m okay with that. I swear it looks like new!