A simpler list of cases

I have to test many cases, but this solution is not very elegant:

if '22' in name:
    x = 'this'
elif '35' in name:
    x = 'that'
elif '2' in name:    # this case should be tested *after* the first one
    x = 'another'
elif '5' in name:
    x = 'one'
# and many other cases

Is there a way to do this sequence of cases with a list?

L = [['22', 'this'], ['35', 'that'], ['2', 'another'], ['5', 'one']]