Short answers, pulled from the story.
Adhiṣṭhāna is a Sanskrit term referring to blessings or inspiration that a Buddhist may receive from a Buddha, bodhisattva, or guru. The word also carries meanings including seat, basis, support, abode, and in a broader sense, government and authority. In East Asian Buddhism it relates to the concept of a Buddha's "other-power."
The Tibetan word for adhiṣṭhāna is byin-rlabs. Scholar Dan Martin argues it traces back to a Chinese term introduced during the earliest arrival of Buddhism into Tibet, likely in the seventh or eighth centuries, and its philologically correct meaning is "received by way of giving." The literal Tibetan has been glossed as "gift wave," though Martin considers this imprecise.
In Shingon Buddhism, adhiṣṭhāna is practiced through rituals involving mantras, mudras, and visualization aimed at achieving honzon kaji, or union with the deity. Minoru Kiyota identifies three kinds in Shingon practice: mudra (finger signs), dharani (secret verses), and yoga through meditation.
The cucumber blessing, called kyuri kaji in Japanese, is an adhiṣṭhāna ritual practiced at Shingon Buddhist temples in summer. Priests and believers pray together to pass the hot season in good health, like fresh cucumbers. Kūkai, the founder of Shingon Buddhism, is said to have originated this practice.
In East Asian Buddhism, adhiṣṭhāna is one of the sources for the concept of a Buddha's "other-power," which is central to Pure Land Buddhism. D. T. Suzuki called the idea one of the Mahayana landmarks in Indian Buddhist history and the beginning of the distinction between other-power (tariki) and self-power (jiriki).
In Tibetan Buddhism, adhiṣṭhāna blessings can be transmitted by a guru during initiation, by the yidam during deity yoga, or by proximity to holy objects such as a stupa or the relics kept inside them. Tsultrim Allione notes that receiving these blessings requires the student to have proper motivation, aspiration, and devotion.