"armour" is a bit general, from the fancy gleaming best tournement suit to the few bits of salvaged chain mail stitched onto an archers leather jacket.
Generally the knights and foot soldiers wouldn't march in anything like full armour, medieval battles were fought on agreed sites when the armies were visible to each other - not ambushes or blitzkreig. So generally you would have a couple of days notice to get your armour on, sharpen your sword and say mass (all equally important to a knight)
Harald Hadrada famously was killed at the battle of Stamford Bridge in 1066 because he had left his favourite chain mail (called emma!) on the ship for safety.
Henry V did order his troops to be in armour for the march from Honfluer to Calais because he did expect an attack at any time from the French and didn't have enough spare scouts or local help to spot an attacking force.
You keep the armour clean with lots of elbow-grease. Squires spent a lot of time cleaning armour, which if you had been fighting in the mud alongside a lot of scared men and scared horses wouldn't have been a pleasant job! If you get the metal clean quickly it won't rust. chain mail was cleaned by rolling it in a barrel of sand. Where do you get barrels of sand on a march? Presumably the supply train contained forges, tools, spare parts, and barrels for rent.
At least late medieval armour steel was 'blued' - you quench the finished steel in oil which coats the surface and protects it. I don't know how long working armour lasted - presumably it depended on how much fighting it's owner did and how scrupulous his squire was. The suits you see in museums and on tombs were special sunday-best sets for show.