Numbers Increase While Letters Decrease

Python 2, 53 52 51 bytes

^{-2 bytes thanks to g.rocket
-1 byte thanks to Jonathan Frech
-1 byte thanks to RootTwo}

def F(x):n=sorted(x);print[n.pop(-(e>x))for e in x]

Try it online!

The sorted list will have the numbers first and then the chars like [3, 5, 6, 'a', 'b', 'x'], then use e>x to filter what is number and what is char, in python any number is less than a list (input) and a list is less than a string.

APL (Dyalog), 27 26 bytes

Expects characters to be uppercase

(⍋⊃¨⊂)@(~e)(⍒⊃¨⊂)@(e←∊∘⎕A)

Try it online!

This is just two applications of the form f@g, apply the function f on the items indicated by g.

For the first application we use:
 f: ⍒⊃¨⊂ the descending grades (⍒) each pick (⊃¨) from the entire argument (⊂).
 g: (e←∊∘⎕A) members (∊) of (∘) the Alphabet (⎕A), and store (←) this function as e.

For the second application we use:
 f: ⍋⊃¨⊂ the ascending grades (⍋) each pick (⊃¨) from the entire argument (⊂).
 g: (~e) not (~) members of the alphabet (e; the function we stored before)

JavaScript (ES6), 71 51 47 bytes

Saved 20 bytes by just using sort(), as suggested by @JustinMariner
Saved 4 more bytes thanks to @CraigAyre

Using a similar approach as Rod's Python answer:

a=>[...a].map(n=>a.sort()[1/n?'shift':'pop']())

Test cases

let f =

a=>[...a].map(n=>a.sort()[1/n?'shift':'pop']())

console.log(JSON.stringify(f(['a', 2, 'b', 1, 'c', 3]))) // -> ['c', 1', 'b', 2, 'a', 3]
console.log(JSON.stringify(f([5, 'a', 'x', 3, 6, 'b']))) // -> [3, 'x', 'b', 5, 6, 'a']
console.log(JSON.stringify(f([3, 2, 1]))) // -> [ 1, 2, 3 ]
console.log(JSON.stringify(f(['a', 'b', 'c']))) // -> [ 'c', 'b', 'a' ]
console.log(JSON.stringify(f([]))) // -> []
console.log(JSON.stringify(f([2, 3, 2, 1]))) // -> [1, 2, 2, 3]