Why did Maniac have mixed feelings about winning the race against Mars Bar in Maniac Magee?

Not only did Maniac win the race against Mars Bar, but he also won it running backwards.  Maniac is glad he won, but he feels bad about humiliating his opponent in the process, and wonders what it is within himself that possessed him to do that.

Maniac didn't want to race Mars Bar to begin with.  The other boy has always been belligerent towards him because he is white, and he only wants to race Maniac because he wants to show him up.  Mars Bar thinks he can beat Maniac because has been working out, and has gotten a fantastic new pair of sneakers.  There are a lot of people watching the contest, friends and neighbors of Mars Bar, so there is a lot at stake.

Maniac is not sure at first how to run the race. 

"Naturally he wanted to win, or at least to do his best...but there were other considerations...whom he was racing against, and where, and what the consequences might be if he won".

Maniac doesn't want to make Mars Bar look bad, but he doesn't want to be beaten either.  Once the race begins, however, he is caught up in the excitement, and, leading all the way, he turns at the last minute and crosses the finish line backwards.  Maniac "regret(s) it instantly", questioning himself as to why he did it.  He is afraid that he might have done it deliberately, disgracing his opponent as a form of payback "for all his nastiness".  In the final analysis, however, "his only recollection (is) a feeling of sheer, joyful exuberance, himself in celebration", doing what he does well with complete abandon (Chapter 38).

What were Maniac's and Mars Bar's emotions after their race in Maniac Magee?

In the book Maniac Magee, the title character beat Mars Bar in a race. In the end, Maniac won the race by running backwards! After they crossed the finish line, Mars Bar was furious. He accused Maniac Magee of cheating and "he shoved [him]" in anger. Maniac realized when he looked at Mars Bar that "the hatred in [his] eyes was no longer for a white kid in the East End; it was for Jeffrey Magee, period."

Maniac Magee was full of regret after winning the race. He looked back and wondered why he had even let himself win. He wondered if he had "done it deliberately, to pay back Mars Bar for all his nastiness."

The crowd that had gathered to observe the race was full of excitement. Maniac was not sure if their enthusiasm was because he had won, or because the race itself had been exciting. This focus on him also made Maniac feel embarrassed. He just wanted to escape from all of the attention from the crowd.

Why is Maniac upset for crossing the finish line backwards in the race with Mars Bar in Maniac Magee?

Maniac is upset with himself for crossing the finish line backwards when racing with Mars Bar because in doing so, he is rubbing his victory in the face of his opponent.  It is bad enough for Mars Bar that he loses to Maniac in a race in which he, Mars Bar, had issued the challenge, but when Maniac crosses the finish line backwards, Mars Bar's loss is accentuated, and he is truly disgraced in front of all his friends.  It's kind of like in baseball, when a person hits a home run and stands there at home plate to admire it while the opposing pitcher stands, embarrassed, on the mound, or in football, when a player makes a touchdown, then slams the ball down and goes into an elaborate dance of victory right there on the field.  Such behavior is not considered to be good sportsmanship, and, at its worst, is insulting to the opposing team.

Maniac had not meant to exhibit unsportsmanlike behavior.  He had simply been caught up in the moment, "feeling...sheer, joyful exuberance, himself in celebration".  Even so, "that didn't make it (an) any less stupid or rotten thing to do".  Maniac holds no ill feeling towards Mars Bar, and he regrets having acted in a way that embarrassed the other boy.  It is true that Mars Bar had been going out of his way to cause trouble for Maniac because, as a white boy, Maniac had dared cross over into the East End of the city, which is inhabited predominantly by black people, but Maniac had still never meant to get back at Mars Bar in this way.  Although there is a moment in which Maniac questions his own motivations, it is generally totally out of character for Maniac to act in a way that is deliberately demeaning to others under any circumstances (Chapter 38).

I believe that you meant to ask why Mars Bar was embarrassed when Maniac crossed the finish line backwards.  I have changed your question to show this.

Mars Bar was embarrassed when Maniac crossed the finish line backwards because it meant that Maniac had beaten him really badly.  Mars Bar, who had racist feelings towards Maniac, had challenged him to a race.  He felt sure that he could beat Maniac.  But then Maniac beat him by such a great distance that he was able to turn around (without meaning to) and cross the finish line backwards.  This was embarrassing for Mars Bar because he lost the race badly in front of all of his friends.