Is praying for the deceased at home the same as praying for them at the grave? And does my personal prayer for the deceased reach them, just as a general prayer for all the deceased in the grave reaches them?

Supplicating for the deceased is legislated, whether at the graveside or away from it. The permissibility of this is indicated by the Prophet's (PBUH) saying: "When a person dies, all his deeds come to an end except for three: ongoing charity, knowledge from which others benefit, or a righteous child who supplicates for him." The permissibility of supplicating for them at the graveside is indicated by the narration that the Prophet (PMPUH) used to visit the people of Al-Baqi' and seek forgiveness for them. There is no difference between supplicating for a single individual exclusively or along with all the deceased in the cemetery. It is better to supplicate for him individually and within the group of the deceased in the cemetery, so as to repeat the supplication for him.