For some reason, Edinburgh isn't generally regarded as one of the great places to visit on a budget. And that probably has to do with the fact that for one thing it is on the British pound, which does immediately put budget travellers at a disadvantage. But also the accommodation here is not cheap, so without a friend to crash with it could indeed be a pricey trip. However, once here there is so very much to do in all seasons that costs nothing or next to nothing that I'm of the opinion that it all evens out very nicely. And the Christmas and New Years celebrations are no exception. Most of the Christmas fun is free of charge for entry, or at least very reasonable. And even though there is admission and some costs for the Royal Bank Hogmanay Street Party, one of the best things Edinburgh offers over the holidays is free, in the 7 Hills Fireworks.
There aren't many places in the city where you can't see them... so no matter what you were doing or where you are at midnight on New Years Eve, just pop outside and watch them. They are a truly spectacular sight, being fired off from the seven hills surrounding the city of Edinburgh and around eight tonnes of fireworks being used each year.
Though this is a wonderful (and free) thing to do and see from anywhere in the city... there are a few prime places that I will recommend. For the best and most amazing view of the fireworks spectacle as a whole there is little better vantage point than from the top of Arthur's Seat or the crags. If you want a memorable New Year's Eve that doesn't involve evening wear, then this is a great way to see in the New Year.
But keep in mind it will be dark and it's quite a hike to make in the day time, as well as the fact that it will be extremely cold and windy up at the top in January. Drinking before hand is definitely not a good idea. But if you are up for that adventurousness of it, I highly recommend it. But if you want something a little less work but with almost as good a view, the view from Calton Hill of the city is hard to beat, and the fireworks seen from here make an amazing sight.
But if these two are still a bit too high, or too much work there is always Inverleith park, which has a wonderful view of the castle, or simply on the Bridges between Princes Street and the Royal mile. There is no traffic because of the street party so you can stand shoulder to shoulder with other locals and visitors (mostly inebriated) and sing in the New Year to the display of fireworks over the castle. But no matter where you decide to watch it from it's a wonderful way to spend the evening.