Differential Equation of a First Order High Pass Filter
I think your equations are largely correct. Going around the loop I obtain, \$U_{in }= Q/C + U_o\$ , just as you did. Differentiate this now with respect to t, to convert the charge, Q, to a current (since \$I=dQ/dt\$): \$dU_{in}/dt = I/C + d U_o/dt\$. Now applying Ohm's law to the resistor, we have that \$I = U_o/R\$. Substitute this in the equation and you obtain \$dU_{in}/dt = U_o/(RC) + d U_o/dt\$. I believe this is the differential equation the question refers to.

Now make the Laplace transform, and the derivatives are replaced by factors of "s", so that
\$s x(s) = y(s)/(RC) + s y(s)\$. The question says \$x(s)\$ should be the Laplace transform of the unit step function, which is \$x(s)=1/s\$. After some rearranging, this gives the result \$y(s)=1/(s + 1/RC)\$. Using a table of Laplace transforms you can take this back to the t-domain (it is one of the standard cases), and obtain \$U_o(t)\$.

Does this seem reasonable?