If the only acceleration is due to the large mass's gravity, and the mass is not exceptionally large, or exceptionally close (i.e. close approach to a black hole or a neutron star), the astronaut will not experience any noticeable acceleration relative to the spacecraft. Gravity affects the spacecraft and the astronaut nearly identically, and the acceleration of one matches the acceleration of the other. This is identical to the situation for a spacecraft in a closed orbit around a planet, which likewise is continuously accelerating towards the planet's center.
The effect of gravity decreases with distance from a large mass, so there's always a gravitational gradient across a spacecraft. If the gravity gradient is steep enough, or the spacecraft very roomy, an astronaut who moves away from the spacecraft's center of mass will find themselves on a diverging orbital trajectory, which will tend to push them further still from the center of mass. This is called tidal force. The effect is too small to be noticeable in any reasonable case: a dangerously close Jupiter flyby, say 10km above the cloud tops, would experience a gravitational gradient less than a ten-millionth of one g-force per meter of difference in altitude. Even in a kilometer-long spacecraft a human wouldn't notice a tidal force from one end to the other.
Because the force of gravity decreases with the square of distance from the center of mass, the gravitational gradient gets steeper as you get closer to the mass; for exotic large masses like neutron stars or black holes, it's possible to get much closer to the mass. In theory you can get close enough for tidal effects to be human-perceptible; in practice you probably don't want to get that close. Spoiler for a well known 1966 science fiction story:
Larry Niven's 1966 story, "Neutron Star" considers the effect of gravitational tidal force across a long, skinny, indestructible spacecraft making a close approach to a neutron star, in a setting where starship pilots have somehow forgotten about the existence of tidal force.
Rocket thrust and atmospheric drag, unlike gravity, act on the structure of the spacecraft directly, altering its trajectory relative to that of the astronaut, causing the astronaut to get squished toward one side of the spacecraft, hopefully the side with padding. Note that, due to the Oberth effect, a close approach to a planet may also be a good time to use rocket thrust to effect trajectory changes, so active thrust may be combined with gravity assist.