No can do.
I was able to find the answer here, written by LIGO scientist Dr. Amber Stuver:
How valid is the wave-like-in-water analogy? Can we “surf” these waves? Are there gravity “peaks” like there are “wells”?
Stuver: Because gravitational waves can travel through matter unchanged, there isn’t a way to surf them or use them for another kind of propulsion. So no gravitational-wave surfing.
The “peaks” and “wells” is an excellent point. Gravity is always attractive because there is no negative mass. We don’t know why but it has never been observed in a lab or any evidence found elsewhere in the universe. So gravity is usually represented on spacetime graphics as being a downward curvature, or your “well.” A mass traveling by the “well” will tend to bend inward toward it; this is gravitational attraction. If you had negative mass, you would have repulsion, which would be represented by a “peak.” A mass moving by a “peak” would tend to bend away from it. So there are“wells” but no “peaks.”
The water analogy is very good at talking about how the strength of the wave decreases as it travels away from its source. A water wave will get smaller and smaller just like a gravitational wave will get weaker and weaker.
Slightly simplified, this means that you can't use gravitational waves for propulsion because they don't transfer energy quite in the same way that water waves do. The analogy breaks down further because gravitational waves are plane waves, not sinusoidal waves - so you shouldn't try to visualize them as being anything like water waves.
