user6312's suggestion to use a tree diagram is good, but I'm a bit
lazy and prefer to do the calculations using linear algebra.
Matrix multiplication automatically adds up the contributions from the branches of the tree diagram.
You start the particle at some state in the set $A=\{ -1, 0, 1\}$. Let $Q$ be the
matrix of transition probabilities from this set into itself, i.e.,
$$ Q=\pmatrix{0&1/4&0\cr 3/4&0&1/4\cr 0&3/4&0}.$$
Then $Q_{ij}^n$, the $(i,j)$th entry in the $n$th power of $Q$, is
the probability that the walk is in state $j$ at time $n$, starting
at state $i$, without leaving the set $A$.
Since you don't want the answer, I will simply say that
the row sums of $Q^4$ give you what you want, each row sum
corresponding to one of the three possible initial states.