It is normal that you might be a bit overwhelmed as a new DBA for your first year. If this situation extends for much longer than that, then something is wrong and it is obviously time to ask for long-term help. This situation is especially true if you are the only DBA or the first Oracle DBA in your organization. Help can come in the form of changing your phone number to non-published within your organization, changing who you directly answer to, or adding another DBA (on-site, remote, or temporary consultant). Compare and evaluate the regular tasks you perform with other DBAs. It doesn't matter if you are only a backup or junior DBA at this point, it is the best time to learn these valuable concepts and still be protected from disrupting the production environment. Everyone wants you to succeed as a DBA because it is a position with a huge amount of responsibility, long hours, and it requires ongoing training to stay competent.
Are you barely able to answer requests as they pour in? Ask that those requests be filtered or first reviewed by a front-line tech support personnel. The exception would be if that is your job—a front-line DBA to serve as part of a tech support organization. On a side note, third-party vendors will put tech people in support positions who give DBA-like advice, but be very wary as they often have little to no actual experience. Remember, a lot of technical helpers only play with databases all day. They don't actually have to manage a live production one with your critical data in it. Learn to say no to these individuals.
A DBA is most often placed underneath the head of IT (most often the CIO) because the database is often central to the entire IT department. If you are one of the many DBAs, your organization will most often have an immediate DBA supervisor that you will answer to first, before the head of IT.
The DBA is not the sole source for any issue related to Oracle, especially if your entire IT department is new to Oracle. Their job is not to train everyone on the basics of Oracle—they don't have the time and probably aren't the best people for that job. Most DBAs don't have the experience of teaching others; they are usually best in a one-on-one situation training a backup or junior DBA. They also shouldn't be subjected to the whims of upper management when there is a technical issue that can't be resolved by the standard front-line personnel, or when the perception is that the issue won't be handled quickly enough.
After a year on the job, you should be proficient enough with My Oracle Support (MOS, Oracle's support site renamed from Metalink) to find out most of the answers on your own and be comfortable enough to implement the solution without help. Using an Internet search engine may be another source of information (beware that the information may not be timely, complete, or even accurate), but it is a source you turn to when other more dependable ones aren't productive.
When you create a MOS Service Request, don't expect an answer that day unless the severity level and contract agreement dictates that level of support. Oracle Engineer support help may be hours or even days from now. Reach out to the Oracle community for help at the same time you enter a Service Request, which often will get a quicker if not better answer than Oracle support, because these people will have encountered the same problem at some point. Not everyone will give reliable advice; learn to recognize those that do. Most often a test case with reproducible results is the best indicator of expertise, or at least a well-thought-out process. E-mail lists and forums have been known in the past for shooting down people who don't read the documentation first. The exception would be if you are at a total loss on how to fix a problem or are experiencing a disaster at a particular moment.