The thing that jumps out to me is more a grammatical thing than a problem with the story - try some different punctuation rather than short sentenc after short sentence. For example -
My alarm clock shrieks and I awake with a jolt, the crust in my eyes from the Sand Man's visit making it hard for me to peel them open. Once I do, I sit up and take in my surroundings - Leo, my cat, is asleep on my favorite blue loveseat by my dark wooden ceder chest. His tiny white paws flick every once in awhile, hinting at the fact that he is probably dreaming. The small rays of sun peak through the gaps in my thin white curtains, dancing around the room as the wind from my open window disturbs their gentle slumber. The air in my room feels crisp and clean. The smell of freshly cut grass fills the air, reminding me that it is now summer, and yet again I have no plans. That realization seems to throw me back onto the bed and beg for more sleep. “No more sleeping in until noon” I remind myself as I pull myself out of bed and slowly waddle like an old woman over to the bathroom.
Note that this allowed me to take out some of your "I"'s, and it sort of flaws. I'm very fond of semi-colons myself. When you can't decide whether you need a new sentence or some kind of continuation, read it as you would say it. An example -
The small rays of sun peak through the gaps in my thin white curtains, dancing around the room as the wind from my open window disturbs their gentle slumber
The comma means a short pause, not long enough to disrupt the flow of the sentence. Now, if I replaced it with a semi-colon...
The small rays of sun peak through the gaps in my thin white curtains; dancing around the room as the wind from my open window disturbs their gentle slumber
That pause is longer, almost like a new sentence but it sticks to the same theme. However, that would work if you wrote it like this-
The small rays of sun peak through the gaps in my thin white curtains; they dance around the room as the wind from my open window disturbs their gentle slumber
Sorry if this seems annoying, but grammar is something that I've worked on in my own writing for a while, and some variation makes everything read better.
Oh, and as for how I threw the hyphen in there, it could easily be replaced by a semi-colon, but it gives some variety. It works in a similar way by connecting similar ideas.
You also need to keep an eye on your spelling - well, which variation of a word you use. In those lines I showed the ;'s is one example - wrong 'peak'. That peak is the top, or a summit. You want 'peek'. Also, 'he stairs at me' - stares.
As for the story itself, it has promise. Your descriptions in the first paragraph, in particular, are very good. One suggestion though is how the phone call ends - I can't imagine any employer hanging up without a farewell or an comment on behalf of the employee. Maybe have her stutter a reply, so as to still show her surprise.
I hope this doesn't put you off - this isn't criticism, it's critique. I look forward to seeing how you go. You definitely have a talent, you just need to polish it. Heck, I've been a 'writer' for years and I still need a lot of work!