For both the #1 most important thing you need is a strong relationship with Christ.
Then....
Minister of Music
Strong communication skills: this is where most of the leaders I've worked for have fallen short. You need to have a vision and be able to convey it.
Organization: Get songs and other information out in a timely manner. Don't call saturday night asking the other musicians and the praise team to have a brand new song ready for sunday morning.
An understanding of the pastor's/church's vision: I was put in charge of a praise team at one church that was only doing old school foot stomping songs when I got there. My vision was to turn them contemporary, however the pastor didn't agree which eventually led to me resigning my position. It was all amicable however a lot of time and energy could have been saved if we had sat down and discussed it all beforehand.
Humility: Being in charge does not make you exempt from the rules. If you expect the singers and musicians to be on time be on time yourself, if you expect them to know songs by rehearsal time, make sure you know them yourself. One of the biggest faults in a lot of leaders is they are great at ignoring the standards they expect those following them to obey. Practice what you preach.
Delegation: Don't be afraid to put others in charge of certain things. My home church has a band leader and his job is to be the liaison between the minister of music and the band, so instead of you having to call every band member about a change you simply tell the band leader and he conveys it to the rest of the band.
Musician
Desire to improve: Too many musicians are comfortable at their skill level and have no desire to grow or improve their skills. In order to be a great musician you have to always be striving to get better.
Structured practice: This is something I'm actually just learning to implement. Too often I'll sit down at the keyboard with my only goal being to learn the new songs for sunday or for my band's next performance. I'm now understanding I need to be more focused in my practice time so now I'm setting a certain amount of time for doing scales(with a metronome), working on songs for church, working on songs for my band, and then an amount of time to explore new techniques, chords, scales, etc.
These are what I feel are some of the most important aspects of being a minister of music and a musician. If you have any questions feel free to leave them here or inbox me.