Yes it is viable but not with Shield.

Your best bet would to use Mage Armor, since it is a single action spell as opposed to the reaction of Shield that has specifics on when you are allowed to cast it (emphasis mine).

Casting Time: 1 reaction, which you take when you are hit by an attack or targeted by the magic missile spell

So unless you actually have someone hit you repeatedly you can't cast this spell all that often per PHB 196. At least by RAW.

A reaction is an instant response to a trigger of some kind...

This section goes on to say that the opportunity attack is one such instance of a reaction but you can't take it unless something would trigger that as detailed in all the descriptions of reactions. This why when you ready an action you are required to specify a trigger.

Mage Armor could be used as often as necessary to regenerate the ward, unless there is errata somewhere that changes the wording to slots as opposed to level. One thing to note is that spells with verbal components could be overheard and thus could give away position or spoil the element of surprise, so choose when to do this with care.

Yes, this would work

The Arcane Ward class feature of the Abjuration Wizard does not specify that the abjuration spell in question has to come from a Wizard spell, nor does it mention spell slots, simply the spell's level, so as long as it's not a cantrip, if you find a way to cast an abjuration spell of 1st level or higher at will (such as by casting mage armor via the Armor of Shadows Eldritch Invocation) then your plan will work. You would be able to repeatedly cast mage armor in this way to restore the ward's hit points to the maximum amount "for free".

From the PHB, page 115:

Arcane Ward

Starting at 2nd level, when you cast an abjuration spell of 1st level or higher, you can simultaneously use a strand of the spell's magic to create a magical ward on yourself that lasts until you finish a long rest. The ward has hit points equal to twice your wizard level + your Intelligence modifier. Whenever you take damage, the ward takes the damage instead. If this damage reduces the ward to 0 hit points, you take any remaining damage.

While the ward has 0 hit points, it can't absorb damage. but its magic remains. Whenever you cast an abjuration spell of 1st level or higher, the ward regains a number of hit points equal to twice the level of the spell.

Without Multiclassing

As pointed out in a comment by @BlueMoon93:

You don't even need to multiclass. If you're a level 18 Wizard, you also have a few at-will spells, and Mage Armor can be one of them

So through the use of the Spell Mastery class feature of the Wizard, you can pick an Abjuration spell as your 1st level spell or your 2nd level spell and cast that at will to restore your Arcane Ward's hit point.

From the PHB, page 115:

Spell Mastery

At 18th level, you have achieved such mastery over certain spells that you can cast them at will. Choose a 1st-level spell and a 2nd-level spell that are in your spellbook. You can cast those spells at their lowest level without expending a spell slot when you have them prepared. [...]