If the domain is given by some parametrization, or implicit curve I know how to find the normal vector but what about $\Omega = \{ (x_1, x_2, x_3) | x_1^2 + x_2^2 <1, \quad 0 < x_3 < x_1 + 2 \}$?
Further, In general domain, is there any systematic ways for obtaining normal vector?
For a surface $X(u,v)$, the normal vector is $X_u\times X_v$, and for the implicit surface $F(x,y)=0$. The normal vector is given as $\nabla F$.
There is no normal vector for a domain. You're talking about a normal vector for the boundary of the domain, and it should point outwards. So, on the bottom disk, it's $(0,0,-1)$. Now, how do you find the outward normal on the (appropriate part of the) plane $x_3=x_1+2$? Then you have to look at the (appropriate part of the) cylindrical part of the boundary, $x^2+y^2=1$.