You can do this by setting the initial condition to nothing:
prob2 = ODEProblem(sys, [:u2 => nothing], tspan)
After which the problem solves without problems:
sol2 = solve(prob2) # yay!
plot(sol2, idxs = [:u1, :u2])
For reference, there is a page on the ModelingToolkit docs about system initialization that contains this specific problem.