This is possible, but only if your product is less complicated than the calculator desktop accessory and you have very experienced automators working on the project. More likely, it will take a few months before automation begins to really pay off. The good news is that once automation begins to pay off, it builds momentum. The organization sees the benefits of automation and may be willing to commit more resources to it. Often, having time saving tools in place means there is more time to create time saving tools.
Test automation becomes a self-supporting cycle.
The bad news is that the first few months are rough. It is difficult to justify commiting talented individuals with strong product knowledge to a long term project when there are short term crises going on now. Clearly communicate the initial goals and milestones of the automation effort to both upper management and the testers. Measure your success against your goals and advertise the results to both upper management and the other testers. No matter how many talented testers you have on staff now, you will probably need help.