Job help!

You need a personal number before getting a job no matter what, unless you can find someone willing to hire you under the table (which is highly unlikely). How long have you got residency here? If you show skatteverket your residency permit, you should be allowed to have a personal number.

Swedish employers pay a hefty sum in employer-fees, insurance, and taxes, so you really need to prove that you are going to be a worthy investment for them. If you don't have residency here and require employment in order to get residency, my best recommendation is to work your butt off and prove to someone that you can be a great employee and get them to vouch for you for residency, albeit that is possibly the absolute hardest way to gain residency, unless you've had a job offer already.