どうしますか "How about ~" is a question on the point of view of JR or a travel agency. It must be 新幹線の(席の)予約をしたいんですが、どうすればいいでしょうか? or like that.
Since で can indicates the means (to transfer in this case), 新幹線で席の予約をする is understandable. However, it sounds like you want to reserve something when you are at a Shinkansen seat, because で usually indicates the location of action.
The seats you can reserve are all 指定席. You can't reserve 自由席, so you don't need to say 指定席.