Assuming I were to decide between getting Benefit A myself or somebody else getting A. All things being equal (need, desire, etc), is it correct to say most ethical theories (other than egoism) would say that I need to let the other person receive the benefit?
[...] is it correct to say most ethical theories (other than egoism) would say that I need to let the other person receive the benefit?
No, because you have not told us enough to apply most ethical theories in the first place.
Let's go through some of the more popular theories:
- Act utilitarianism is indifferent between the two courses of action you describe, assuming A has no effect on any other person and there are no other courses of action available.
- Kantian ethics would want to know whether you or the other person has any preexisting right to A. If not, it might have further questions regarding A, particularly involving where it came from, whether anyone else has any rights to it, and so on.
- Rule utilitarianism, contractualism, and a number of other ethical theories would want at least as much information as Kantian ethics, if not more. For example, some variations of contractualism would have questions about whether the benefit A arises in other contexts, so that we might generalize it to a larger set of people, and then make decisions behind a veil of ignorance (as Rawls describes).
- Virtue ethics would likely praise your decision to give A to the other person as an example of generosity or (perhaps) self-sacrifice. However, absent further information, it's unclear whether it would object to your decision to keep A for yourself. This might be a case of greed, or you might have legitimate reasons for wanting A and denying it to the other person.
- There are a number of non-realist metaethical theories, which deny that right and wrong exist (or make other metaethical assertions that render this question unanswerable). We can ignore those for the sake of this discussion, but you should know that they exist.
A very common fallacy is to believe that ethics is altruism. Wrong.
First, ethics prioritize the group. Not the individual. Example: probably in all ethical systems it is ethical to save your family before you from death. So, if an ethical decision is to be made, that depends on the option individual-group, the group outcome is the most ethical one.
Second, ethics are not incoherent with the individual behavior. Example: it is ethical to help other getting water in a drought situation. But if the individual goals are contrary to the group goals (e.g. you save water for yourself, causing other to die, or you kill someone in your own interests), they are unethical.
So, the rule is this:
If the individual goals are coherent with the group goals, any behavior is ethical, and vice versa.
Before applying the rule, consider that group goals are usually long-term, and individual goals could be short-term (example, selling cocaine seems to generate money, jobs and produces pleasure to a lot of people, but only in the short term; in the long term, group survival is menaced).
Regarding the situation you mention: you have to decide between you and an individual in the group, that means your interests are contrary to those of the group. The behavior is unethical. Even if it implies dying. Example: Gandhi, he didn't feared getting beaten or die, everything for a quite abstract ideal. In this case, Gandhi could be compared to his killer, who, probably in the same economic, social, educative situation, preferred "receiving the benefit", to paraphrase you.