When washing one's hands in the morning, one should abide by all the requirements that apply to Netilat Yadayim before eating bread. This means washing with a utensil from which one pours from his hands onto each other, and using clean water that had not previously been used for any purpose. The utensil should be complete, without any holes.
It appears from the Shulhan Aruch that although one preferably should meet all these requirements when washing his hands in the morning, they are not indispensable for the performance of the Misva. Thus, it seems, according to the Shulhan Aruch, one who does not have a utensil for washing in the morning may run his hands under the faucet, and may even recite a Beracha when washing in this fashion. Since a utensil is not strictly required, one fulfills his requirement even without a utensil, and so one who washes without a cup can recite a Beracha.
However, Hacham Ovadia Yosef ruled that since many disagreed with the Shulhan Aruch on this point, and maintained that one cannot fulfill the obligation of the morning washing without using a utensil, we must apply the rule of "Safek Berachot Le'hakel" – that a Beracha should not be recited in situations of doubt. The Aruch Ha'shulhan (Rav Yechiel Michel Epstein, 1829-1908) challenges the Shulhan Aruch's position, noting that both the Rambam and the Rashba (Rav Shlomo Ben Aderet, Spain, 1235-1310) rule stringently, that the morning hand washing requires all the conditions that are required when washing before a meal. Therefore, given the uncertainty surrounding the possibility of fulfilling the obligation without a utensil, one who does not have a utensil should wash his hands, but without reciting the Beracha of "Al Netilat Yadayim." This was the view also of Rav Haim Palachi (Turkey, 1788-1868). Hacham Ben Sion Abba Shaul (Jerusalem, 1924-1998) ruled that one who washes Netilat Yadayim in the morning without a utensil recites a blessing, but Hacham Ovadia disagreed with this position, as discussed.
The Yalkut Yosef writes that if one washes from a faucet without a utensil, he should place his hands under the water three times in alternating fashion, just as one normally pours water over his hands three times in alternating fashion.
If a person washes in the morning without a cup, he should wash again later once he has access to a cup. If a long time has passed since his first washing, then he does not recite the Beracha over the second washing. If, however, he has access to a cup not long after he had washed straight from a faucet, then he recites a Beracha over the second washing. Although he might have fulfilled the obligation through his initial washing, nevertheless, he may in this case rely on the opinions who allow reciting the Beracha of "Al Netilat Yadayim" after the morning hand-washing. As we saw in a previous installment, some communities had the practice of reciting "Al Netilat Yadayim" together with the morning blessings in the synagogue. Although we do not follow this practice, one may rely on this opinion if he washes without a utensil and then soon thereafter is able to wash with a utensil.
Summary: One must wash his hands in the morning with a utensil, just as one must when washing before eating bread. However, if one does not have a utensil for the morning hand-washing, then he washes his hands by placing them under the faucet three times in alternating fashion, but without reciting the Beracha of "Al Netilat Yadayim." If soon thereafter he has access to a cup, he should wash again with a cup and recite the Beracha. If he is able to wash from a cup only after a considerable amount of time has passed since the initial washing, then he washes again but without reciting a Beracha.