When reviewing job postings asking for 1-2 years experience, often the best job posting is an old job posting. A job posting that is several weeks old may provide your best opportunity for positive exposure. Remember, companies often place job postings to replace employees who have left the company. Employees who have left were no longer entry level when they left (even if they began as entry level). They were experienced employees, so the company wants to replace them with other experienced workers. That is why job postings are rare for entry level jobs.
Old job postings are an excellent source of employer information.
So if after two or three weeks the employer is still unsuccessful in replacing the person with an experienced person in the field, it may be time for the employer to reconsider how it will fill the position. The departing employee will have already left (giving two weeks notice) and the employer may be open to considering other options. Entry level may be under consideration for the very first time in the process. Since very few are looking at the old job postings (newer job postings get the highest search results, your competition is next to nil.
The best approach is to approach the employer directly, not responding to the old job posting, but targeting your resume and cover letter toward summarizing your qualifications for filling the particular job. State that you heard that there might be a need for a _____ and that you will soon be available. If the position is still open and they are considering other levels, you may have an entry point.