How to numerically solve a reaction network with fast equilibrium steps?
For A adsorption:
$$r_1=0$$ 
because the reaction is in equilibrium.So,
$$K_1=\frac{[A]}{P_A \times [*]}$$
From this equation you can relate $[A]$ to $[*]$.Similarly, using the last reaction you can relate $[B]$ to $[*]$. Your [A] and [B] are connected via reaction 2.
$$\frac{d[A]}{dt}=-\frac{d[B]}{dt}=-k_1[A]+k_{-1}[B]$$
This is a first order differential equation, so you need only one boundary condition. You have three unknown and three boundary conditions $([A]_0=[B]_0=0$ and $[*]=1.00)$.