Hey now, don't judge me. I told you this week was going to be hectic. :P
What modern cultural issues are closest to this deity?
This is all my own personal feelings so this may different from what is out there. I think things like education is a big deal for Odin. He is a deity of wisdom and brought the runes, so knowledge is something near and dear to to him. The lack of good schools or omitting knowledge or even perpetuating misinformation about something would bother him.
War would be another major thing for Odin. While being a deity of battle, I do think that even he would tire of it and fighting over such menial things would be against what he strives for. He would know that war is an evil necessity( in some cases).
I'm trying to think of other issues that could be in Odin's sphere of effect. Perhaps the future of wolves, due to two companions. Maybe even keeping the wild horses roaming free and healthy due to his connection to Sleipnir. The state of schools in the world. Mead making, because, well, he brought us mead!
There are so many aspects to Odin that picking a few is seeming to be a little difficult.
Has worship of this deity changed in modern times?
I would have to say yes. The sacrificing of humans has stopped, as far as I know, and animal sacrifices have probably come to crawl but I do believe there are some that do so. Since I haven't found any notes of other ways of worship to Odin, I can only speculate after the sacrifices. We still sacrifice but not in the same way. We offer items to Odin and pledge to him, but probably in a different manner. Who truly knows?