Eg=Pg*Δ t=J^2*Δ t/σ
I assume it's a typo since you have the correct expression on the next line.
I have two problems with this derivation. I can't finish your problem because I don't know the answer to the second.
My first objection is your notation:
Eg/ρ is the energy density<-- Yes.
But you said
ΔT = Eg/s
which is not true.
ΔT = [(Eg/ρ)*ρ]/s = [total energy input]/[specific heat]
(Unless "s" is the specific heat density??)
The second objection is that you are assuming that ALL the energy due to the current flow is absorbed by the material. I find this assumption to be realistically flawed unless ALL of the current's energy is absorbed by the material: ie. we are speaking of a perfect insulator. (In which case σ = 0, which is obviously not going to be true since that would cause ΔT to go to infinity.)
However I will note that I suspect your derivation is intended to be the correct one because the equation you are trying to derive ( Δ T=(J^2* Δ t)/(s* σ* ρ) ) makes no mention of the material's thermal reaction to a current, as it would need to if my objection were correct.