I believe I know what formula to use; I just don't know what I'm missing for the X variables.
I believe the relevant formulas are:
$$s_f^2 = s^2\bigg{[}1+\frac{1}{n}+\frac{(X-\bar{X})^2}{(n-1)(s_x^2)}\bigg{]}$$
where
$$s^2 = .1265^2 = .0160023$$ $$n = 62$$
and $$\hat{Y} \pm t_cs_f$$
where
$$t_c = 2.915$$ & $$\hat{Y} = 1.2130$$
If this is all correct, I just need to figure out the X terms in the first formula. I can't seem to work out how that's possible with the given information.
The slope coefficient is $\hat{\beta}_1$ in $Y = \beta_0 + \beta_1 X $, so its $99\%$ CI is $$ \left( 1.21 - 0.1265t_{0.995}(60), 1.21 + 0.1265t_{0.995}(60) \right) $$