The Slutsky equation links Hicksian and Marshallian demand functions.
Hicksian demand minimizes the cost necessary to reach a certain utility. In your question, you've labeled this $x_1^s$ (although i'm not familiar with this notation). Because Hicksian demand holds utility constant, it measures the pure substitution effect.
Marshallian demand maximizes utility given a fixed income. In your question, you've labeled this $x_1^m$. Because Marshallian demand holds income constant, price affects Marshallian demand both because a price increase for one good makes the other good more attractive (substitution effect) and because it decreases the different kinds of good baskets you can buy (income effect).
Note that, in order to find $x_1^* = 2m/3p_1$ and $x_2^* = m/3p_2$, you maximized utility given a fixed income, by solving $MU_{x_1}/MU_{x_2} = p_1/p_2$ with $m=p_1x_1+p_2x_2$. And so, what you've found and labeled $x_1^*$ and $x_2^*$ are Marshallian demand.
That takes us back to the Slutsky equation
$$\frac{\partial x_1^s}{\partial p_1} = \frac{\partial x_1^m}{\partial p_1}+x_1\frac{\partial x_1^m}{\partial m}$$
What is this saying? Remember, Hicksian demand is pure substitution effect. So $\frac{\partial x_1^s}{\partial p_1}$ is the substitution effect. And Marshallian demand includes the substitution effect and the income effect.
So what this is saying is that the substitution effect ($\frac{\partial x_1^s}{\partial p_1}$) is what you get if you start with the total effect ($\frac{\partial x_1^m}{\partial p_1}$), including income and substitution effects, and then take out the income effect ($x_1\frac{\partial x_1^m}{\partial m}$).[note below]
So where does that leave you?
You can calculate the income effect as $x_1\frac{\partial x_1^m}{\partial m}$. In other words, as the price rises, it rises for every one of the $x_1$ units you're buying, and so the effect on income is the price change (1) times the number of units ($x_1$). And income affects your demand for $x_1$ through $\frac{\partial x_1^m}{\partial m}$, so the income effect is $x_1\frac{\partial x_1^m}{\partial m}$. Since you have Marshallian demand, just take the derivative of $x_1^m$ with respect to $m$ to get $\frac{\partial x_1^m}{\partial m}$, and then multiply by $x_1$ to get the income effect.
You can calculate the substitution effect, then, in one of two ways. You can use the Slutsky equation: calculate the total effect $\frac{\partial x_1^m}{\partial p_1}$ by taking the derivative of $x_1^m$ with respect to $p_1$, and then plug that into the Slutsky equation with the income effect to get the substitution effect, $\frac{\partial x_1^s}{\partial p_1}$.
Or, you can calculate Hicksian demand $x_1^s$ directly by solving $MU_{x_1}/MU_{x_2} = p_1/p_2$ with $U(x_1,x_2) = s$. Then, take the derivative of $x_1^s$ with respect to $p_1$ to get the substitution effect.
[Note: It might seem weird to "take out" the income effect by adding it. But remember that the total effect is negative, since a higher $p_1$ leads to less $x_1$. And so adding on a positive income effect (higher income leads to more consumption of normal goods) is indeed "taking out" the income effect.]
