How does the taylor rule make the MP curve upward sloping? I know it's because $\bar n \tilde Y$  is added in but isn't the MP curve chosen by the government?
In effect, the desired real interest rate $R_t$ is chosen by the government (via changes in the nominal interest rate). But it is chosen based in the Taylor rule. The "standard" IS-MP-PC model (as in Jones's Macroeconomics textbook), is as follows:

IS:

$$\tilde{Y}_t = \bar{a} - \bar{b}\left(R_t - \bar{r}\right)$$

PC:

$$\Delta \pi_t = \bar{v}\tilde{Y}_t + \bar{o}$$

MP (Taylor rule):

$$R_t = \bar{r} + \bar{m}(\pi_t - \bar{\pi})$$

In the latter equation, the central bank/government choses $R_t$ according to deviations in inflation from the target. As inflation increases, the real interest rate increases. 

Now, according to the Phillips Curve, inflation responds positively to demand shocks ($\tilde{Y}_t$). In consequence, the real interest rate responds positively to $\tilde{Y}_t$. This is why, in the IS-MP diagram (with axis $R_t$ and $\tilde{Y}_t$), the MP has a positive slope.