Is it necessary to raise the hands when supplicating for someone's success or guidance in ordinary situations?

Raising the hands during supplication (du'a) is a Sunnah, not an obligation. It is one of the means by which supplications are answered, and it is better than supplicating verbally without raising the hands. If you say to someone: "May Allah grant you success," or "May Allah protect you," or "May Allah guide you," it is better to raise your hands during this supplication. However, if you do not do so, there is no sin upon you.