近いうちにmeans "these day" "the coming day", お目に掛かりたいmeans that I want to meet you. I want to meet you and talk with you. 敬語の表現ですので、友達には"早く会って、お話ししたいと思います"の意味です。 "存じます"は"思います"の敬語です。

I'm not an expert of my language, so please wait for the correct answer. I think it's an expression. This うちに means like "in / within" in English. (I think there's a difference between "within" and "during / between" in English and you use one of them dependimg on the expression, though I'm not sure...) I hope this was helpful to understand it a bit.