How to determine the temperature dependence of equilibrium constants? Given $K_c=2.8\times10^2$ at $727~^\circ\mathrm{C}$ for 
  $$\ce{ 2SO2 + O2 &lt;=&gt; 2SO3},$$
  calculate $K_c$ and $K_p$ when 8 grams of $\ce{SO2}$ result in 0.32 grams
  of $\ce{O2}$ forming at equilibrium (temperature is not $727~^\circ\mathrm{C}$).


Now at first I thought this was straightforward :


determine $K_c$ at this temperature from the equilibrium conditions
apply Van 't Hoff equation to determine the temperature 
calculate $K_p$ from $K_c$ and the temperature


However right after that part of the question the next part asks 


  Is the new temperature higher or lower than $727~^\circ\mathrm{C}$?


This implies that it is possible to calculate $K_p$ without calculating the temperature? With the new $K$ values then provide a qualitative argument for the temperature increasing or decreasing without having to calculate it?
I agree that the second question seems a bit leading, but personally I would just calculate all quantities of interest as you laid on in your plan. The van't Hoff equation requires the reaction enthalpy change, so I assume using a tabulated value is fair game. If so, you will know whether the forward reaction is endo- or exothermic. If you calculate the equilibrium constant at the unknown temperature as you planned, you can compare it to the equilibrium constant given in the problem (for 727 C). Comparing these two equilibrium constants will tell you which way the reaction proceeded, and with the sign of the enthalpy, you should be able to deduce whether heat was added or lost... i.e. whether the new temperature is higher or lower than the initial state, 727 C (without having to go through the entire calculation).