First! The correct spelling here is actually 楽しみにしております. To perhaps make things more clear, the root phrase here is 楽しみにする, though I don't think I've ever heard someone omit the お in front of 楽しみ. The phrase your friend gave you is the humble equivalent of お楽しみにしています (おる = いる in 謙譲語｛けんじょうご}), which is often appropriate when writing letters so let's go with that.
He told me that in Japanese you don't really say 'I hope this' or 'I hope that'
Right. It's uncommon to say things like "I wish" or "I hope" in regard to actions dependent on another's volition; typically one uses phrases like those (ように願う, ように祈る) when speaking about things that can broadly be described as determined by "fate" (getting a job, meeting the love of your life, getting an A on a test), but you wouldn't say them directly to the people responsible for determining them. So when you say:
僕は連絡を出来ますと楽しみおしていますでも、お好きなよしてください。
It's inherently going to sound a little off because that's not how one would typically express such a desire in Japanese/Japanese culture. Before we get to how best to say what you mean, let's clean this up a bit:
僕は
Perfectly fine
連絡を出来ますと
First, we tend to use が when using the potential form of a verb (連絡ができる). Second, if we were using a verb such as 願う or 祈る we'd use V-potentialように, but 楽しみにする uses を as a particle.
でも
けど and が are generally your best bet for sentence-ending conjunctions when looking to express "but." が is a bit more formal, so let's go with that.
お好きなよしてください。
Almost! Remember that よ = ! and よう = way. As such, it should be お好きなように
We end up with something along the lines of:
僕は[相手の名前]が次に連絡できることを楽しみにしておりますが、お好きなようにしてください。
Here we run into the issue of cultural sensitivity (for lack of a better term). If you say explicitly that you're looking forward to the next time that they'll be able to contact you, you apply some pressure to the recipient to make it so that they have the ability to correspond soon. This isn't the worst thing in the world, but I think it'd sound more natural to say something along the lines of "I look forward to our next opportunity to correspond" or また連絡できる機会をお楽しみにしております because it takes the focus from your partner's ability to reply and places it on an indefinite future opportunity. Similarly, saying お好きなようにしてください afterwards sounds a little ツンデレ, like "I can't wait until we talk about, but I do whatever you want because it's whatever." I'd probably just end on the お楽しみ bit, but you could add something like 慌てて/急いで答える必要はありません if you really wanted to drive home that they don't have to rush. But again, you're writing a letter in 2017 so I think the lack of speed is known to be endemic to the technology.
また連絡できる機会を楽しみにしております[が、慌てて答える必要はありません。]
