The ward begins at maximum hit-points. The first version of the PHB stated the same passage as:

The ward has hit points equal to twice your wizard level +your Intelligence modifier.

I believe the change in text was made to address questions about the maximum, but in doing so I guess it introduced this question.

1. Can a 5e simulacrum gain temporary hit points YES
2. Can a 5e simulacrum regenerate an arcane ward YES

Simulacrum PHB p276

It appears to be the same as the original, but it has half the creature’s hit point maximum and is formed without any equipment. Otherwise, the illusion uses all the statistics of the creature it duplicates.

The simulacrum lacks the ability to learn or become more powerful, so it never increases its level or other abilities, nor can it regain expended spell slots.

If the simulacrum is damaged, you can repair it in an alchemical laboratory, using rare herbs and minerals worth 100 gp per hit point it regains. The simulacrum lasts until it drops to 0 hit points, at which point it reverts to snow and melts instantly.

The spell description about learning or becoming more powerful refers to implicit power i.e. level, training, hit points, spell slots. It does not refer to being the recipient of the effects from the use of spells or abilities that do not restore lost resources. Bless, Guidance, Enhance Ability, raging, Gauntlets of Ogre Power etc. all work on it. A healing potion does not. A Pearl of Power does not, and, to look at a different Wizard Tradition, Expert Divination (PHB p116) does not restore spell slots (RAW, though the actual description of the ability might lead to a house rule that the spell slot is not regained but actually represents it not having been used in the first place).

Temporary hit points PHB p198

Some spells and special abilities confer temporary hit points to a creature. Temporary hit points aren’t actual hit points; they are a buffer against damage, a pool of hit points that protect you from injury.

Temporary hit points "aren’t actual hit points" so gaining them does not represent recovering hit points in any way, they are an effect of a spell or ability so the simulacrum can gain the benefit, just as they can wear Gauntlets of Ogre power to give them 19 strength.

Arcane Ward PHB p115

you can weave magic around yourself for protection. 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.

The protection gained from this ability is a ward that has it's strength measured in hit points. The simulacrum does not gain or regain hit points with this ability, it gains a ward, an ability's effect, where "Whenever you take damage, the ward takes the damage instead" with it's effect measured in hit points, and it is the the ward that regains strength when the simulacrum casts certain spells.

As an aside, the ward's strength is not temporary hit points so the Simulacrum, or any other creature with this ability, could have those at the same time too.