If Hyper Synchron was used for a Synchro Summon of a Dragon-Synchro monster and Light-Imprisoning Mirror is face-up on the field at the same time, will the effect to gain 800 Attack and Remove from play at the End Phase be negated by Light-Imprisoning Mirror?

No because it is a lingering effect.

I thought Hyper Synchron effect type is a 'Trigger' effect type? Usually 'Trigger' effect gets negated by Light-Imprisoning Mirror?

The gain attack part is Trigger, then after that remove from play is Lingering.

So if Mirror open before the synchro, then it negates the whole thing altogether. But if after the synchro, and the attack already gain liao then you activate Mirror, then by rights the lingering effect would still let it be removed from play.

I may be wrong though, so don't take my word for it until someone can find a link to the specific ruling.