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.
I'm not certain how I should interpret this phrase regarding class abilities that automatically generate an HP buffer. If a Fiend Patron Warlock's Simulacrum reduces an enemy to 0 hit points would it gain the temp hp as described in the class feature? If an Abjuration Wizard's Simulacrum casts an abjuration spell would the arcane ward recover hp?
On one hand, you could say that passively regenerating resources goes against the intent of a non-regenerating simulacrum. If it gains temporary hp without spending resources of its own then it is "becoming more powerful."
On the other hand you could say that the class features are part of the creature's natural abilities, so the simulacrum should be able to do it. Gaining the temporary hp isn't making the simulacrum more powerful, it's just utilizing the ability that it already has.
Best Answer
Simulacrum PHB p276
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
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
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.