$ 800.00
That's 2602.08 zloty according to google.com/search?q=dollar+zloty
I don't know the Warsawian rental market but I can look on the internet for such a place if you give me a better idea of where exactly.
I know you said downtown, does that mean the old town? (I know the "old town" was reconstructed from paintings but for the sake of simplicity that's what I will call it.
I know from the other forum that you are not very computer literate but if you'd like I could do a test run, give me an approximate area (including an exact name) and I will do a search for you and show you the results just to set your mind at ease.
I believe this process should be really up to you as on the internet your anonymity would be compromised and I would like you to move in without the whole internet knowing.
I see no problem finding a place for you for that price but I live in krakow and although I presume things are more expensive in Warsaw they can't be that much more.
The only thing is, it is easter, I have much to do and I will only be able to help you next week.
Edit*
I am using a different computer and the cursor seems to jump.... bugger.