Maximized tax revenue & foreign labor
I give here the procedure but with price $p=1$ you can do it yourself without that simplification.
Set up Lagrange
$$\mathcal L(c,l,\lambda) = u(c,l) - \lambda (c-\bar w(1-t)(24-l)) $$
clearly
$$\frac{\partial \mathcal L }{\partial c} = \frac{\partial u}{\partial c} - \lambda = 1- \lambda$$
so $\lambda=1$ and constraint is binding. Therefore
$$c^\star = \bar w(1-t)(24-l^\star)$$
Furthermore
$$\frac{\partial L}{\partial l} = (24-l)^{1/\eta} - \lambda \bar w (1-t) = 0$$
since $\lambda = 1$ it follows that
$$l^\star = 24 - (\bar w(1-t))^\eta$$
plugging $l^\star$ and $c^\star$ into utility function, starting with $c^\star$ you get
$$\bar w(1-t)(24 - l^\star) - \frac{\eta}{\eta + 1}(24 - l^\star)^{\frac{\eta + 1}{\eta}}$$
then $l^\star$ to get
$$\bar w(1-t)[\bar w(1-t)]^{\eta} -  \frac{\eta}{\eta + 1}(\bar w(1-t))^{\frac{\eta(\eta + 1)}{\eta}},$$
reduce this to get
$$(\bar w (1-t))^{\eta +1}[1/(\eta + 1)]$$
insert optimal tax and get
$$\left(\bar w \frac{\eta}{\eta + 1}\right)^{\eta +1}[1/(\eta + 1)]$$