Answer
Increasing
Not bounded
Work Step by Step
We have:
$$\begin{aligned}
a_{n+1}-a_n&=(n+1)^3-3(n+1)+3-n^3+3n-3\\
&=n^3+3n^2+3n+1-3n-3-n^3+3n\\
&=3n^2+3n-2\\
&=3n(n+1)-2
\end{aligned}$$
As $n\geq 1$, we have
$$3n(n+1)-2\geq 4>0.$$
So $a_{n+1}>a_n$, therefore the sequence is increasing and not bounded.
Check on a graph.
