The earliest you can begin working on an H-1B visa is October 2014. Your employer will need to hire an immigration attorney to prepare the case. I recommend you start right away, as the first week of April is likely to be the latest date you could file and have a chance at approval.
No one will be able to calculate your chance of success without discussing the case with your employer.
Talk to the company's immigration lawyer. They are paid to answer your questions. Hire an immigration attorney of your choice for second opinion.
The information contained in this answer is provided for informational purposes only, and should not be construed as legal advice on any subject matter,not should it be viewed as establishing an attorney client relationship of any kind.