Learn Arabic (Modern Standard): نحن نتذكره.

It is attached directly to the end of the verb! The suffix ـه (-hu) means "him". In Arabic, you don't use a separate word for an object pronoun; instead, you stick it right onto the end of the action.

If the verb نتذكر already starts with نـ for "we", do we really need to include the word نحن?

No, it is not strictly necessary. Because the نـ prefix on the verb already tells us "we" are doing the action, dropping نحن is very common. Including it here just adds a little bit of emphasis, like saying "We remember him."

Could نتذكره also mean "We remember it"?

Yes, absolutely! Arabic doesn't have a neutral "it" pronoun. All nouns are either masculine or feminine. The suffix ـه (-hu) is used for masculine people ("him"), but it is also used for grammatically masculine objects and concepts ("it").

How do you pronounce the transition between the verb and the attached pronoun?

In Modern Standard Arabic, a standard present tense verb like نتذكر usually ends with a "u" sound (a damma). So the verb itself is pronounced "natadhakkaru". When you attach the ـه (-hu) pronoun, you keep that vowel, making the whole word "natadhakkaruhu".