first thing, you wrote . this is not correct because is not a field. for example, (1,1,1,1,1,1,1,1) is the unit in this ring, but then there is no inverse for (0,0,0,0,0,0,0,1).

when you want to find the inverse for an element f(x) in a field where F is a field and p(x) irreducible you use the euclidean algorithm to find the h(x)=gcd(f(x),p(x)) and the a(x), b(x) that satisfy a(x)p(x)+b(x)f(x) = h(x). because p is irreducible you must have h(x) is invertible so

in your case we have

some rearranging ...

so the inverse of f(x) = x^7+x+1 is x^7. (notice that because the field is F2 minus and plus are the same.)