I'll address storage of antimatter, because that is the one thing in your question humans have done successfully so far. While we may someday build an antimatter-based propulsion device, it's a ways off. Storing antimatter is another story.
Currently, the best way to store antimatter is a Penning trap. It uses a magnetic field and an electric field to store charged particles (so this would not work for neutral particles unless they were "attached" to charged particles, perhaps like neutrons are bound together with protons in a nucleus). A magnetic field (or an electric field) on its own could not keep a particle in a stable position because of Earnshaw's theorem; using a Penning trap provides a loophole.
Now, a Penning trap is generally used for keeping a particle in a desired position, i.e. in a static state. There is a chance you could refine one to move some of the antimatter, but it would be difficult. That said, any civilization that can build a starship probably has pretty sophisticated technology. . .
Okay, I might as well try to address one more problem that is . . . well, a big problem. When you bring the antimatter into contact with matter, you can't simply have it in the storage area. If it's in a small storage area, the energy released may destroy the Penning trap (or whatever else you're using). If it's in a large storage area, the explosion probably won't be near whichever end of the craft you designate the rear. Either way, the explosion won't be directed rearwards, as with a typical rocket.
The solution like this might be to accelerate the antimatter and matter out the end of the spaceship. Particle accelerators do this via superconducting magnets. The problem is, these accelerators are incredibly large - the Large Hadron Collider is 27 kilometers in circumference! Perhaps that would be tough to do on a small spaceship. To solve that issue, I would suggest using a small ion engine: to accelerate the matter and antimatter. Have them accelerated perpendicular to each other and away from the ship, and you could direct the explosion to the rear. Again, the particles have to be charged, but it's a small price to pay for a powerful spacecraft.
This is my first activity on the Worldbuilding site; I hope my answer suits it.