没关系，一有消息我就马上告诉你，请问用英语怎么翻译？

看这是在什么情况下用了，楼上的那些答案都不错。
没关系可以用: Don't worry about it, No problem, Relax, Don't sweat it, It's okay...等方法来表达，（基本上可以说，从前往后，最正式到最随意，根据情况 Calm down也可以放在“没关系”的位置）
马上通知的话，I will let you know (inform you) the instant I got the news.

不过一般不这么用，最常用的是“我会尽快通知你” I will inform you ASAP (as soon as possible).
你会是第一个听到的 you would be the first person to know.
想文艺点的话不妨试试: I will deliver the news to you faster than Hermes himself.

P.S. Hermes - 希腊神话里的信使。

That's OK. Once I get news, I will inform you at the first time.

纯属个人见解。
如果满意，请记得采纳，(*^__^*) 谢谢~~

That's OK. I'll keep you informed.

Don't worry, I will inform you the moment I got the news.

祝你好运~_~

It doesn't matter, I would inform you once I have got the news.