Is there a keyboard shortcut in Word for typing an arrow symbol?

You need to use the Numpad (the other numbers won't work!).

↑   ALT+24  Upwards arrow
↓   ALT+25  Downwards arrow
→   ALT+26  Rightwards arrow
←   ALT+27  Leftwards arrow
↔   ALT+29  Left right arrow
▲   ALT+30  Upwards filled arrow
▼   ALT+31  Downwards filled arrow
►   ALT+16  Rightwards filled arrow
◄   ALT+17  Leftwards filled arrow

People who type -> probably don't know that if you type --> you get the default arrow in Word.

If you actually want --> to be shown in Word, you can type --> and press backspace, which will undo the automatical conversion to the arrow image.

You could try just using alt codes of these characters:

• Left Alt+26 for →
• Left Alt+27 for ←