Brother, Umrah can be done anytime except for the days of Hajj. Getting Umrah visa is an issue as Saudi Arabia now focuses on Hajjis and process their visa applications in these two months. Check with visa agent and if you could get visa, you can go for Umrah. In my personal experience one day in Mecca and Medina is not enough. I love to spend more time so my advice is to plan your trip properly.

You can go by air or car but by air is more convenient. Finding parking in holy cities is not easy and can lot of time. Pls ensure that you need to tell the visa agent up front as visa type for air and land are different and cannot be interchanged. There are few bus options too in Qatar which I heard are also good.

Hope the above info would help you. May Allah accept your prayers.