You can get the best answer by calling the benefits support center at 888-SBUX-BEN. They can look up your info and tell you exactly when you might be able to get insurance. However, it's unlikely that you'd be able to get coverage in the next 2-3 months. If you need to make plans for surgery soon, you might have to look at other options for insurance on
$1. Health insurance is horrendously complicated so if you're new to the process, you might want to do some research and watch some explainer videos
$1, or ask someone you know and trust to help you.
Once you have insurance, the insurance company decides what to cover. (Starbucks deals with many different insurance companies, and your options will be different depending on where you live.) Any of them will probably cover your surgery **if** it's deemed medically necessary, but the cost you end up paying can vary a lot, which is why it helps to do research
In general, for Starbucks benefits, you become eligible by working 240 hours over 3 full consecutive months. After that, you get a letter in the mail inviting you to enroll, but they'll give you a deadline. If you miss the deadline, you can't do anything until the company-wide enrollment period every August, and at that point you wouldn't actually have insurance until October. (If you call the benefits center, they can tell you where you stand here.)
Once you schedule your surgery, it's best to request time off at least 3 weeks in advance, just like if you were going on vacation. But if you don't get time off approved in advance, you're still allowed to miss work for necessary doctor visits, which this would obviously be.
Good luck!