Your problem can be split in two parts, getting there and surviving the trip.
Most "exotic" propulsion systems require lots of electronics for control, or just are based in electric effects themselves. Going with 2019 tech level, what's in rutine use today for commercial spaceships should be your best bet.
Most ships use liquid fuel rocket engines. Those are efficient for the long range, but they have a few caveats for your mission: They require fine control of the burn if you dont want them to blow up, and the massive ammounts of fuel and oxidiser require pretty big pumps, which will need to be electric in a spaceship. The other problem with this approach is that this kind of engine doesn't like reignitions, and you will probably need to do a lot of correction burns along the trip.
Your next option is the hypergolic, it is less efficient and also needs pumping, but at least the control is very simple and can be done via mechanical means. You can do the pumping by having compressed gas push the liquids; won't do for the big trans-martian injection burn and the capture but would probably be OK for the correction burns along the trip.
For the big burn, I would go with the scariest of all: solid boosters. Those can be ignited via a hypergolic mix, and once they start burning there's no control needed because there's no way to control them; they burn until expended.
So, using a staged approach, I'd have a solid first stage for the trans-martian injection, followed by a hypergolic second stage for mid course corrections and a final solid stage for the mars capture.
As you won't have a control computer or anything that can compensate for asymetric trust, you'll need to have one of each, perfectly aligned with the center of your ship.
You'll also need to calculate your burns as precisely as possible. The most critical is the trans-martian injection one, it is the longest and also the one that has a bigger potential to screw your whole mission if done wrong. As you will be starting for an already known orbit around Earth, it is also the easier to calculate; you can even have it precalculated using a computer if that's allowed by the rules.
Your engines will need to be throttled down so that they burn slowly during a long time. A pilot seating in the front of the ship will use an optical finder to align the ship with a star and steer it if it spins off course; the slower the burn the easier it will be to align the ship.
For the attitude control, you could also have hypergolic thrusters like most spaceships do, but unless perfectly balanced, they will also propel you off course. A flywheel on a gimbal is the perfect option. Your 24 crewmembers can put their muscles to the work; orient the flyweel in the right direction, start spinning it and watch your ship counterrotate until it finds the right orientation.
The navigation shouldn't be that difficult, but it will need a lot of slider rule time. Observe the planets, find them in precalcualted tables and figure out how well you are doing. Correct your course accordingly.
The easier first: Don't bother with a recycling system, just bring enough water and low-residue food to last the trip. You will need to keep it aboard, though. Anything you expel from the ship will either alter your course if you do it with some force, or just stick around if you dont, surrounding you in a cloud of residue that will cloud your windows and difficult all your maneuvers. It would also alter the ship mass in unpredictable ways, so better bring it along.
For warmth you can just do like Mark Watney and bring a few plutonium pellets, they will radiate enough heat to keep your ship at a reasonable temperature. You don't want to be too warm to avoid people sweating, the water is better kept inside their bodies.
Another consideration is clothing. You won't be able to wash it so you'll do as the ISS and bring enough clean clothes for all the crew.
Air quality will be trickier; you can do as old spaceship did and have a pure O2 atmosphere, but you'll need to get rid of excess CO2, and have something that does that efficently; I personally have no idea how to do that without electricity and avoiding outgassing.
Finally, you'll have 24 people in crammed, smelly qarters for 9 months. You better figure out how to keep them entertained without electronics.