I wrote this under the assumtion that the screen was placed at the plane where the rays come to focus. I now realize that the question does not state that. If the screen is not at the focal point, then no, the rays will not have equal travel time.
Your intuition is correct, each ray would reach the screen at the same time. In fact, in some sense, you could define the image plane of any imaging system as the plane where all rays from the object plane have an equal travel time. That would be a bit of a sloppy definition and it isn't quite complete in the technical sense, but for an intuitive explanation it will work in most cases.
Speaking in terms of geometrical optics, optical engineers usually refer to the optical path length (OPL) rather than the travel time of the light, but they are equivalent quantities, differing only by a factor of the speed of light (c). The OPL in a medium of refractive index $n$ is defined as:
Where $L$ is the actual distance traveled in the medium. So, for a ray passing through several diferent materials (like some air, a lens, and then air again) the total OPL can be expressed:
This quantity should be equal for each ray when it is properly focused from the object point to the image point.
This makes intuitive sense when you think about it. A lens with positive power is thicker at the center than the edges, and typical glass has $n\approx1.5$. The actual length of the ray through the center of the lens is shorter than the rays that go through the edges of the lens, because it follows a more direct path; but the central ray also goes through the thicker part of the lens, which increases its OPL to match that of the rays through the edge.
That pretty much sums up the geometrical optics picture.