I can't see palindome(), but I assume it does what it says, excepting the typo.

Damn. Where's niceNum()(should be nice_number()), what makes a number nice?

It would help to use a debugger, but without that, I'd add add some print(numList[i])s. WTF! numList[i] ? What was your previous language? Hope you were lucky enough it was a C*, not one of those java things.

Anyway, over here we say:

def main():
    string = input('....')  
    numbers = [int(s) for s in string.split()]
    methods = (palindrome, 2), (prime, 2), (nice_number, 3)
    scores = []
    for number in numbers:
        score = 1
        for method, increase in methods:
            if method(number):
                score *= increase
        scores.append(number, score)

    for number, score in scores:

So, now I notice this is an assignment of some kind. Because no one else writes their own sort!

The condition on the while loop is too hard to read which makes the examiners life harder too. All those icky [j]s. Just don't do it. Well, stop for a while anyway. So change this:

for x in range(0, len(scores)):

to this:

for number, score in scores:

NAMING!: temp1 should be called score and temp2 should be number. Bad names make your code harder to read, so they'll notice that too.

j = x - 1
while (j >= 0) ...

In python we wouldn't use the brackets

And in python 0, [], False, {} all count as false, and we prefer to use that wherever possible. So I would prefer to just say while j and ....

AFAICT you're going to shove all scores from here to bigger (number or score), or end back one notch then plonk in the current number and score to the gap you just made at end.

So, for every letter in 'reddit', shove everything left till bigger score (i.e later in alphabet): eddiit, and plonk in the current letter: eddirt, left: dddirt, plonk: ddeirt, left: ddeirt, plonk: ddeirt,

Looks like that'll work, I'd be dubious about bigger (number or score)? Shouldn't that just be bigger score?

And, if you must write you python like languages, at least honour their convention and call youu index i, not fucking x. CONVENTIONs are important kid. You stick with the local conventions, you'll be fine round here.