Should I wait to upskill myself or contact professor about the PhD opening asap?
I don't think you stand to gain much by waiting. Especially if, as you state in your question, the position might no longer be available within two months.

Either the supervisor is open to working with you and you'll have the opportunity to get to know him a bit and to demonstrate your ability to find your way in his field of research, or he's looking for a PhD student with a prior skillset you're unlikely to acquire in 1-2 months time anyway. 

If he's up for it, I'd say it's always a good idea to have a general talk with him to get to know your prospective field better, show your interest and to get targeted feedback on whether your Masters might simply be good enough in the first place. That doesn't exempt you from reading up a bit about his current research and it's background though.