Assuming this is not a cap-exempt H1b (non-profit or on a school campus), an H1b effectively can only be applied for every April 1 for a start date of October 1. (There is a quota on how many are available each fiscal year. Fiscal years start in October 1. They can apply 6 months before the start date, so the earliest that they can apply is April 1; and in recent years, they fill up within a few days in early April, so it pretty much must be applied by April 1.)
However, you must meet the requirements for H1b at the time it is filed. You said you will graduate in April, so you may not have the degree by April 1 this year, and so may not be able to use that degree to qualify to apply this year. If you have already completed all the requirements for the degree, you might be able to use it even if you haven't received the actual diploma.