B1 to H1B Help Needed !

I have an approved I797B for my H1B Visa which was processed in premium Processing but it does not have my I-94 attached to it.

Since i was already in US on B1 status from my previous employer, my new employer applied for COS from B1 to H1B

Is the above process even legitimate? If yes then please anyone tell me procedure how cos from B1 to H1B can be achieved after i have received my I-797B/I-129 approval notice?

The time it takes to complete this process? I have round about 3-4 months left for my exit and I-94 expiration.