Answer
a) $\rm Down, \;3\; cm$.
b) $\rm3.5\; cm$
Work Step by Step
a) The water level in can 2 rises from 5 cm to 6.5 cm which means that whether can 1 moved up or can 2 moved down, but by what distance?
We know that the water volume in our system, the two cans, is constant. So if it is raised in can 2 by 1.5 cm, it must be lowered in can 1 by the same amount since the two cans are identical and have the same cross-sectional area $A$ز
This means that the second can moved 3 cm down.
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b)
The volume in both cans initially is
$$V=V_1+V_2=A_1h_1+A_2h_2$$
where initially $h_1=h_2=5$ cm,
We also know that the two cans are identical, then $A_1=A_2$.
so that
$$V= A h +A h =2Ah_i\tag 1$$
Now we have the same water volume
$$V=2\color{red}{\bf\not} Ah_i=\color{red}{\bf\not} Ah_{1f}+\color{red}{\bf\not} Ah_{2f}$$
Now we know that the final height in can 2 is 6.5 cm, so
$$2h_i=h_{1f}+h_{2f}$$
$$h_{2f}=2h_i-h_{1f}=2(5)-6.5 =\color{red}{\bf 3.5}\;\rm cm$$
