First, pray - ask God to show you what to do. He does care about the details of your life. Next, take a deep breath - always remember to breathe -sounds funny but we get in to a panic and stop regular breathing.
Now - information is power - so gather the information you need. It takes MONTHS to be evicted. There's no way your landlord can throw you out if you're behind only on this month's rent. In NY it's 3 months of *no* payment before they can even begin the process - and it's the 6 month point before an eviction notice is made - at the earliest. but if you can show you're making a good faith effort to pay the rent and can show partial payments they won't even begin the proceedings. Call your town or county office and find out what the eviction process is where you live. Having that information will get you out of your anxiety pit.
IMPORTANT - as a few other posts suggested - talk to your landlord. We used to have an apartment in the upstairs of our house and when our tenant was between jobs or when his car broke down he was late or paid what he could and eventually caught up. He always called us before, or the day the rent was due. Your landlord wants the rent - of course - but he or she would also rather have the apartment filled, knowing that some portion of the rent is coming in and that it will get caught up. The other option for them is that the apartment stays vacant until he can get someone moved in. He's not getting any rent during that time - that's not what they want.
Ask to make a payment plan with your landlord tell him how much you'll be able to repay with each paycheck after you get yoru first paycheck. Open, honest discussion is always the thing to do. Tell him you are determined to catch up and make him whole, that you love your apartment, etc. The more he feels you're committed to doing the right thing the more willing he'll be to do the right thing for you.