Assumptions on preference relation
The preference relation is not transitive. As a counterexample take $x=(0,0), y=(0,1), z=(1,1)$.
For monotonicity let $x=(x_1,x_2)=(y_1+1, y_2)$ so that the $x$ bundle contains more of some commodity but no less of any, than $y=(y_1,y_2)$. Then
$$ (y_1+1+y_2-y_1-y_2) = 1 \leq 1 $$
so that the preference relation is not monotonic.
so that the preference relation is strictly monotonic.
The preference relation is continuous since
$$ \{x| x\succeq y\} = \{x_1, x_2 | x_1 + x_2 - y_1 - y_2 \geq -1\} $$
is closed and
$$ \{x| x\preceq y\} = \{x_1, x_2 | x_1 + x_2 - y_1 - y_2 \leq 1\} $$
is also closed
Finally, it is not locally non satiated. Take any $\epsilon\leq 1$ as an example.
EDIT: It is not strictly monotonic, see the comment below