The crew would feel as they accelerate by a constant 9.8 m/s^2. They would not feel any slow down in time. For an external observer, the crew would never reach speed of light, but they would close the gap continuously.
If you could somehow indefinitely maintain 9.8m/s^2 acceleration, as you get really really close to c, your definition of a “second” becomes so long that it basically goes to infinity (time starts to travel infinitely slow for you.) Maintaining 9.8m/s^2 at such velocities starts to require infinite energy. If you were somehow able to reach c, your internal clock would be so slow (and the rest of the universe would be going by so quickly) that would experience the end of the universe essentially instantaneously.
So the practical answer to what the crew would experience on the 355th day of 1G acceleration is: “That’s impossible.” The energy requirements approach infinity, and so does the speed of time for the rest of the universe they’d see out their window.
This description doesn't seem quite right to me. The laws of physics work the same in all reference frames, so from the frame of the crew, day 355 would simply be yet another day of 1G acceleration. At least inside the spacecraft, they would notice no difference from day 354 (or any other previous day).
When they look out the windows, they will see odd things with respect to the apparent speed of other visible objects (including they never get farther/closer at a rate faster than C). But there's nothing theoretical that prevents you from continuing to accelerate at 1G forever.
Edit: I guess what I don't like about your description is that you mix reference frames. A spaceship can maintain 9.8m/s^2 (from its perspective) infinitely with constant energy expenditure. Energy expenditure rises asymptotically only if you require constant acceleration of the spaceship from Earth's perspective. But that's a peculiar way of framing it.
I think they would never reach the end of the 354th day, because they'd be caught in something akin to Zeno's Paradox: the closer they got to c, the more time would slow down for them.
You could draw up a table mapping shipboard times to Earth times: the length of time passing on Earth between each onboard second/millisecond/microsecond etc. would get longer and longer, with the final moment at which the ship achieves c never actually arriving.
ETA: basically what @ninkendo said (but they said it better, and first).
