I suspect we'll get E1 only AFTER rearrangement. It's not likely to be straight elimination, because, as Dan pointed out, there are no beta-protons. It's unlikely to be SN1 because primary carbocations are too unstable to form in any appreciable amount. For the same reason, it's unlikely to form the carbocation, then rearrange, then eliminate. It's unlikely to be SN2 because of the severe steric hindrance of beta branching. I just lectured on beta branching this morning, and neopentyl electrophiles react some 100,000 times slower than ethyl bromide in SN2 reactions. So if you let it go long enough, yeah, you'd probably get some SN2... eventually.
So I predict one of the rare cases where rearrangement and LG expusion occur simultaneously. This gives the stable tertiary carbocation. We're in good base, so E1 elimination should occur after carbocation formation. That's my guess. My sketch of the arrow-pushing is below.
Below that is a scheme from the alcohols chapter of Brown Foote and Iverson's Organic text. It shows precedence for this type of rearrangement/carbocation formation. That system is in weak base (actually acidic conditions, so net SN1 results), but the theory is the same?