If you can catch a cheap flight to Dublin, take a Ryanair out to Berlin and catch one back to Dublin from Scandinavia. Overland from Dublin would cost out the wazoo in time and money.
Are you a student? I just got what I think is a good deal out to Heathrow in the height of summer... $585 JFK - LHR from StudentUniverse.com.
Including airfare, I think you could easily keep it under $2000. If you spend $75/day that makes $1050.
You might want to check out a three country
select pass for train travel.
Never been to Oslo or Copenhagen but you CANNOT skip Berlin!!!
I am not a fan of sleeping on the train. I'm sure you have heard the advantages, but the disadvantage is that either you will be sleeping in a sitting position or you will shell out the equivalent of a hostel bed for a bed on the train. Yes it can save you time but I prefer to explore a city well rested.
Basically just sleep in hostels or CouchSurf, cook your own food most of the time, and don't go wild with the alcohol.