I shot Bullseye several years ago. One of the things you need to get ready is...your shooting arm. I presume that you know that Bullseye is shot one-handed. It is important that you have enough strength in your grip and arm to be able to hold your arm out for a few seconds at a time. It's surprising how heavy that gun can get when holding it out there getting the shot to break.
You need to dry fire your .45 a bunch to practice your hold. I have a spot on the wall in the basement that I sight on to practice. Snap caps are good but not absolutely necessary. You shouldn't dry fire your .22 unless you use some kind of snap cap to protect the firing pin and the edge of the chamber.
Don't go out on the town the night before and have too much training.
Extra magazines are helpful because you only have a little bit of time between rounds. (the term escapes me at the moment for what each individual course of fire is.)
If you enjoy shooting Bullseye, you'll probably want to look into a simple reloading set-up so that your ammo expenses won't eat you up buying factory ammo. You can also load lighter, such as 3.5 grains of Bullseye Powder pushing a 185 grain bullet. It's a nice light load that gets the job done.
I noticed that you only speak of one gun for each of you. You do know that a standard Bullseye match has a rimfire, centerfire, and .45 relays. You can use your .45 for the centerfire as well as the .45, as most people do. You can't use your .22 for any match except for the .22 relay. So...you're going to need 2 more guns if you're going to shoot the whole match together. Or, one of you shoots in the morning and one shoots in the afternoon.
A gun case is necessary to carry your supplies to the line. Mine also has a spotting scope attached to it. Good hearing protection is necessary and probably required by the range. Good shooting glasses protect your eyes as there is going to brass headed your way unless you're on the far left side of the line.
Make sure your gun handling etiquette is spot on as you will be told, none too politely, to correct what you're doing wrong such as sweeping the line with the muzzle, heading out to your target before the RSO/match director declares the line safe. If you're not shooting, the gun has to be on the bench, magazine out, slide back(cylinder open), muzzle downrange. I've seen people escorted off the property for not strickly following the rules and for good reason. When a cease fire is declared, no matter when, the rules above apply. Your match director will tell everyone what is going on. When he is satisfied that all is well, he/she will state "ready on the right, ready on the left, ready on the firing line....fire."
I hope that this helps. Hope that if you already knew all of that, you forgive me for presuming otherwise.
Have fun, shoot lots of practice rounds and put 'em the X ring!