Growing up with a brother was an appalling experience for me; my brother was a burden.
It was not completely appalling since you gained your good qualities from your brother's bad qualities or troublesome antics, so maybe substitute the word appalling.
Also, instead of saying burden, try substituting with troublesome or handful for better context.
Many people would say that growing up with a sibling was an enjoyable experience. I would say the opposite.
Many people do??? I also have a troublesome brother, so I find the statement funny!
I became more responsible because I had to take responsibility of my family, especially my brother and I became more patient.
The last statement should direct toward your roommate. Remember that this is a note to your roommate. You can directly state that you will show these qualities toward your roommate somehow.
Overall, I like the message you are trying to get across!