what I can do from know till then to prepare for the trip?

You can't go past Lone Pine Lake without a permit, so the first thing you need to do is get a permit if you don't already have one. Whitney Zone permits are among the most difficult to get in the country and are given out via a lottery held every March. If you don't have a permit yet, you can still get one by checking for cancellations on the recreation.gov website or showing up at Lone Pine a day early and hope for a walk-in permit. You have to get lucky though.

The second thing you have to do is to learn how to handle yourself at altitude if you don't already know. Learn what altitude sickness is, how to prevent it, and what to do if you get it. If you've never been above 12,000' before and you're not one of the lucky few who are minimally affected by altitude, you need to take steps to prepare yourself. Altitude is the main reason hikers have to turn around on Whitney.

The third thing you have to do is learn how to check weather in the mountains if you don't already know how. Weather in the mountains isn't like weather near the coast. It can be highly variable and fast changing. Bad weather or not being prepared for the weather is another common reason hikers turn around on Whitney.

The fourth thing you have to do is learn some basic snow travel skills if you don't already know how. The upper parts of the main trail will have snow and ice in June and will almost certainly be treacherous without the right skills and gear.

The fifth thing you need to do is train. If you're already in decent shape, focus more on strength work than aerobic work. Do low weight at high reps. Body weight squats and lunges are good. Stairs help too. When do your training hikes, focus on intensity more than distance. It's better to do a 10 mile hike that gains 4000' than a 20 mile hike that has the same elevation gain but isn't as steep. Try to find trails that gain 700' per mile or more because that's what you'll face on Whitney.