Hi and welcome to the Dubai forums winnie
Yes, you need your UAE residency visa before you can sponsor a maid. There are other conditions - your salary must be more than AED 6000/month, or AED 5000/month if compnay provides you with accommodation. Bachelors cannot sponsor a maid, and the maid has to be a national of Bangladesh, Ethiopia, India, Indonesia, Philippines, or Sri Lanka. The maid visa is renewable annually and costs AED 5100 plus a few other fees (allow AED 1000). There is also a deposit of AED 5000 required but I'm not sure if that's only for the entry permit for the maid, or also for the residence visa which you apply for after the maid has entered the country. Check a website like dubai.ae or dnrd.gov.ae for more details. How long it takes I'm not sure but as the entry permit for the maid is probably valid for 30 days, I would assume that the system should result in the residency visa issued within that time, if all documents are in order. There could be an extra fee you could pay for urgent processing.
How long it takes for your own residency visa is hard to say. It depends mostly on the company (although they'll often tell you otherwise). Some people get their residence visa within a couple of weeks, for others it takes months because the company is slow. Sometimes there can be delays with the UAE immigration department - usually around holidays or Ramadan, or possibly because the company is in their bad books for some reason. Yes, as far as I know you can pay a fee for quicker processing. Here's more about UAE visas