A word for 'hiding strong emotions'

Solution 1:

Stoic is a term for someone who can handle pain and hardship without showing one's feelings or complaining

Solution 2:

It sounds like your partner is reserved. From Oxford Dictionaries:

Slow to reveal emotion or opinions

This fits your description of your partner as someone who generally keeps his emotions hidden, but who does share them with you (though, I'm guessing, only after a gradual process of getting to know you). Note that the term reserved doesn't imply a lack of emotions or opinions. In fact, the definition from Collins Dictionary is

Someone who . . . keeps their feelings hidden

which (in addition to sounding very much like your description) strongly implies that there are emotions to be hidden. However, a person who is reserved could be mistaken as unemotional by people who don't see beneath the surface calm.

For your example sentence, you need the noun form.¹ From Cambridge Dictionaries:

reserve noun (SHY BEHAVIOUR)
★ [ U ] tending to keep your feelings or thoughts private rather than showing them

So you could say

His reserve enabled him to hide his strong emotions from others.

You could also add an adjective to make the sentence more pointed, as in his natural reserve or his habitual reserve. Similarly, if you wanted to stick with the adjective, you could rephrase slightly:

His reserved (nature/demeanor/etc.) enabled him to hide his strong emotions from others

Or

Because he was (always/naturally/habitually) reserved, he was able to hide his strong emotions from others

¹_{Of course there is also the noun form reservation, but that generally isn't used with this sense of reserve and reserved: When a table is reserved, someone has a reservation; but when a person is reserved, that person only has reserve.}