A clever way to solve this kind of problems (with a Cobb-Douglas function) is as follows:
$ax_1^{a-1}x_2^{1-a} - \lambda p_1 = 0$
$x_1^a(1-a)x_2^{-a} - \lambda p_2 = 0$
Bringing the terms involving $\lambda$ to the RHS:
$ax_1^{a-1}x_2^{1-a} = \lambda p_1 $
$x_1^a(1-a)x_2^{-a} = \lambda p_2 $
Dividing the first equation by the second equation:
$\frac{ax_1^{a-1}x_2^{1-a}}{x_1^a(1-a)x_2^{-a}}=\frac{\lambda p_1}{\lambda p_2}$
$\lambda $ can be cancelled. In addition you can cancel out $x_1^{\alpha}$ and $x_2^{-\alpha}$.
$\frac{ax_1^{-1}x_2^{1}}{(1-a)}=\frac{ p_1}{ p_2}$
$\frac{ax_2^{1}}{(1-a)x_1^{1}}=\frac{ p_1}{ p_2}$
The exponents are not needed anymore.
$\frac{ax_2^{}}{(1-a)x_1^{}}=\frac{ p_1}{ p_2}$
$\color{blue}{***}$
Solving for $x_2p_2$
$x_2p_2=\frac{p_1(1-\alpha)}{\alpha}x_1$
Inserting the term for $x_2p_2$ in the budget restriction:
$y=x_1p_1+p_2x_2$
$y=x_1p_1+\frac{p_1(1-\alpha)}{\alpha}x_1$
Factoring out $x_1$
$y=x_1\left( p_1+\frac{p_1(1-\alpha)}{\alpha} \right)\Rightarrow y=x_1p_1+\frac{x_1p_1}{\alpha} -x_1\frac{p_1\alpha}{\alpha} $
The third term is the negative of the first term:
$y=\frac{x_1p_1}{\alpha} \Rightarrow x_1=\alpha\frac{y}{p_1} $
For a given y and a given $p_2$ the maschallian demand function is $x_1(p_1,\overline{p}_2,\overline{y})=\alpha\frac{\overline y}{p_1} $
The same can be done for $x_2(p_2,\overline{p}_1,\overline{y})$. Repeat the similar steps after $\color{blue}{***}$.
In (c) you are asked to calculate the income and substitution effects for a discrete change in the price of good $1$ from $1$ to $3$. Thus you will not be calculating the effects using derivatives.
In (a) which demand functions were you asked to find? Just the Marshallian, or the Hicksian (compensated) demand function as well?
The Hicksian demand function (found by minimizing expenditure subject to attaining some level of utility $u$) is: $$x_1^h(p_1,p_2,u)=\left(\frac{2up_2}{p_1}\right)^{1/3}\qquad x_2^h(p_1,p_2,u)=\left(\frac{up_1^2}{4p_2^2}\right)^{1/3}.$$
At the income $m=12$ and initial prices (both $\$1$) the Marshallian demand functions you derived tell you that consumption of good $1$ is $8$ units and consumption of good $2$ is $4$ units. Thus initially utility is $U(8,4)=256$. After the rise in price of good $1$ to $\$3$ consumption of good $1$ is $8/3$ units. The total effect of the price rise on consumption of good $1$ is $$x_1^m(1,1,12)-x_1^m(3,1,12)=8-8/3=16/3.$$ The substitution effect can be found by considering the effect of the price change on consumption of good 1 keeping utility constant at each its initial level. This is where the Hicksian demand function is needed. The Hicksian demand for good $1$ at the new prices and old utility is $$x_1^h(3,1,256)=\left(\frac{2(256)1}{3}\right)^{1/3}=\frac{8}{3^{1/3}}.$$
It follows that the substitution effect is
$$x_1^m(1,1,12)-x_1^h(3,1,256)=8-\frac{8}{3^{1/3}}\approx 2.45$$
while the income effect is
$$x_1^h(3,1,256)-x_1^m(3,1,12)=\frac{8}{3^{1/3}}-8/3\approx 2.88.$$
Of course you could do that the other way and calculate the substitution effect by keeping utility constant at the new level (which corresponds to doing what I did above, but for a fall in price of good $1$ from $3$ to $1$). This would give you a slightly different answer (for marginal changes in the price there is not this problem of two different ways of calculating the substitution effect).
In terms of answering (d): have you seen a discrete version of the Slutsky equation?
Best Answer
Suppose that $w_1^1$ and $w_2^1$ denote the endowments of agent $1$. And $x_1$ and $x_2$ the corresponding demands. Then the budget restriction for this agent is
$$p_1x_1+p_2x_2\leq p_1w_1^1+p_2w_2^1$$
Next we use the information about the endowments of agent 1: $w^1=(3,3)$
$$p_1x_1+p_2x_2\leq p_1\cdot 3+p_2\cdot 3$$
At the first interval no demand for good 2. That means that $x_2=0$.
$$p_1x_1\leq p_1\cdot 3+p_2\cdot 3\Rightarrow x_1\leq \frac{p_1\cdot 3+p_2\cdot 3}{p_1}$$
Agent $1$ want $x_1$ only (no $x_2$) if $\Large{\frac{\frac{\partial U1}{\partial x_1}}{\frac{\partial U_1}{\partial x_2}}}>\frac{p_1}{p_2}\normalsize\Rightarrow \frac{1}{2}>\frac{p_1}{p_2}\Rightarrow p_2>2p_1$. This is the first part of the interval. I think you can deduce the other intervals.