I agree that the part about recharging is a bit ambiguous, because it's part of a two-sentence paragraph:
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.
If these were written in the other order, I don't think there'd be any potential for confusion, but this way, I can see someone reading the second sentence as conditional on the first. But, I don't think it's meant to be: if that were the case, the second sentence should be more clear about that.
So, I'd read it as your two spells (4th level and 5th level) recharging the ward to 18hp. And, this is also kind of sloppily worded, but presumably the ward's initial hit points are also the maximum — you can't "recharge" beyond that.
This reading is confirmed as the design intent in a tweet by co-designer Jeremy Crawford — thanks to wax_eagle for checking.
Note that the ward does not mention temporary hit points. It is a stand-alone rule, and it should be read just on what it says. Even if the ward acts similarly to having temporary HP in some ways, it doesn't say that they are, so they aren't — and can work differently in some of the details. (And, crucially, you might have both the ward and temporary hit points.)
For what it's worth, this does seem pretty awesome: a 10th level barbarian with a 20 con has 125 hit points. A level 10 abjurer with a 16 con has 72, and a 25 hp buffer — and if she uses all of her spell slots casting abjuration spells, that can recharge up to 82, for an effective total of 179. If one of those spells is Stoneskin, she has the same resistance to normal weapon damage as the ranging barbarian, although probably not as high an AC without making a lot of the castings Shield.
On the other hand, a 10th-level Moon Druid (say, around 83 hp) can shift into Ankylosaurus form twice between short rests, giving what's effectively two 68hp buffers and also you are a dinosaur. I expect once we get into Deep 5e Optimization Territory, there will be plenty of other things which outstrip the arcane ward.
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.
Best Answer
The ward takes the damage first regardless of what order they apply
The following Q/A establishes that damage to your temporary hit points still counts as you taking damage and with that knowledge, there are two possible cases:
Is a concentration check required when temporary hit points absorb all the damage?
The ward clause applies first; in this case you are no longer taking damage so the temporary hit points won't do anything.
The temporary hit points clause applies first; in this case you are taking damage so the ward's clause will now apply, taking that damage instead.
This is also supported by lead game designer Jeremy Crawford in an unofficial ruling (a tweet):
It's different if you believe temporary hit points taking damage doesn't count as you taking damage
This stance is taken by the second answer to the initially linked question and in this case, the order will matter because either clause applying will result in you no longer taking damage.
In this case I would default to the optional rule from Xanathar's Guide to Everything for resolving simultaneous effects:
Either way, you never make a concentration check
Note that under either belief, no matter what, a concentration check will not be made. In the first account the ward is always taking the damage, not you; in the second account either the ward of the temporary hit points take the damage and neither would count as you taking damage.