Interesting ques.
Well, e1 proceeds via carbocation. So, any reaction which favours formation of carbocation will favour e1
e.g: Resonance stabilised cations(benzylic, allylic, or even cyclopropyl ring stabilised)
polar protic solvent
high temp.
Now, e2 proceeds via transition state. So, factors whiich stabilise the TS help e2
So, less substituted 1 degree carbon will give e2. But here, substitution may get preferred in gas phase or non polar/ polar aprotic solvent
In your ques, 2 degree alkyl bromide is given. So, we will expect e1 with rearrangement