Equilibrium constant in terms of partial pressures
However, surely $P_j$ alone is dimensionless as it's a partial pressure defined by dividing a pressure by a pressure - leaving no units. How does that work? Or am I wrong by assuming that a partial pressure has no units?


You are wrong to assume that partial pressure has no units.  Partial pressure has units of pressure, such as mmHg, atm, bar, psi, etc.

For ideal gas mixtures, $y_j=\frac{P_j}{P_{tot}}$ or equivalently $y_j P_{tot}=P_j$, i.e. the partial pressure $P_j$ of a gas is equal to its mole fraction $y_j$ times the total pressure $P_{tot}$.

The expression you gave for $K_p$ is dimensionless because $P^\Theta$ has the same dimension as $P_j$.


  Do I assume that $\frac{P_C}{P^Θ}=1$....


That part sounds good.


  leaving me with $K_p=\frac{P_{CO}}{(P_{O_2})^{1/2}}\frac{1}{P^\Theta}$


Not sure about that one.  How is $K_p$ dimensionless in what you wrote?