Can a first degree carbocation form if a methyl shift is possible?
The cation is produced when a methyl group migrate from C-2 to C-1 as the C-Br bond is broken. The simultaneous migration of alkyl group and departure of leaving group to form a tertiary cation  is therefore faster and more favourable than formation of primary cation due to simple loss of leaving group. 

The products observed are those that result from further reaction of the so formed tertiary cation. The mechanism mentioned in the pic you've attached is incorrect, 1° carbocation intermediate is not formed at all. 

Several comments correctly mention that formation of primary carbocation is not favourable. So, it is important to note that methyl shift and departure of leaving group in this case are simultaneous processes - primary carbocation is not formed at all! 

P.S. 
Also, I'd like to quote some evidence I found for a very similar reaction of 1-bromo 2,2-dimethyl propane: 



Above is the solvolysis reaction of 1-bromo 2,2-dimethyl propane. 

Source: Reactive Intermediates in Organic Chemistry: Structure, Mechanism, and Reactions - By Maya Shankar Singh (Wiley Publishers)