This is one of the verses in the Bible that is grossly misinterpreted.
What the verse really is saying is that if you have to say you are more spiritual than another in order to obtain a position or an opportunity in church — not business, or profession, church only— you should abstain from such an action. Rather, the verse advises, allow the person who already has put himself or herself forward to have the position or opportunity.
The verse has nothing to do with looking out for others before yourself. In so far as that is concerned, consider that Hebrews 12:2 states:
Let us fix our eyes on Jesus, the author and perfecter of our faith, who for the joy set before him endured the cross, scorning its shame, and sat down at the right hand of the throne of God.
In Philippians 2:4, Paul reiterates,
Each of you should look not only to your own interests, but also to the interests of others.
Christians are not called to abandon their own interests. Rather, they are called not to be willing to mistreat others in pursuit of their own specific interests. In the decision not to mistreat others, and accompanying decision to treat others right we look not only to our own interests, but also to interests of others.
Jesus sacrificed for you and I because a reward already was promised before ever the sacrifice was necessary. If we are to sacrifice as Christians, we must have some understanding of what exactly we expect the reward to be, and the source of the reward. This is the standard set by Jesus Himself.