First locate the zone of interest (marina, business bay, downtown, greens, JVC, JLT etc.) and then go through as many ads on the websites you mentioned as you can in that area to get a feeling of the offer in the area and the prices. Next, go to a number of viewings and talk to the agents as they know a lot and they won't tell you anything unless you ask. Watch some youtube videos about Dubai expats and their experiences with renting an apartment.
About the rent - cash is preferred only if you'll be paying the full year in advance.
If you want to split it to 2, 3 or 4 (or even 6) payments - checks are preferred and not many landlords will accept that.
Same thing goes for monthly payment - not many landlords will accept that as well.
Do not sign a tenancy contract unless you're sure that's the place you want to spend next year.
Terminating tenancy contract is difficult and will be at your expense.
Have some money on the side since you'll have to pay a couple of deposits (for eg. deposit for DEWA is about 2.5k AED) and also the agent commissions are about 1-2k AED.
BTW: I have one good 1BR apartment to rent. Brand new, never lived in. You have my email above