I'm sorry but that doesn't seem entirely correct. Whilst the inverse may be hard to find, the value of the inverse function cannot be found be subbing 1 into the normal function. Whilst the substituion method may not be able to be utilised, the graphical analysis method may.
I'm sure you know that the inverse of a function is the function reflected over the line y = x. Either you can make a rough sketch by hand or use a graphical calcultor (common in most schools nowadays for senior mathematics courses).
By sketching the graph of f(x) = x + cosx, it can be seen that it will have an inverese function due to the graph being 1:1 (meaning that each x point has only one corresponding y point and vice versa).
Another aspect of inverse functions is that essentially the x and y values swap for the inverse. This means that for the numbers subbed in (if y = 1, x = 0) on the inverse graph there will be a point (0,1) or at x = 0, y = 1.
Taking this into account by finding the value of the original function when x = 1, the value for the inverse function when y = 1 will be found.
so y = x + cosx
if x = 1, y = 1 + cos 1 (your calculator will need to be in radians mode to calculate this properly)
from this you get y ≈ 1.54
this means that the value of f^-1(1) ≈ 1.54