The hardest part is documentation. 1945 is perhaps easier to "create" an identity following the war, as so many we displaced. But after that it would mean needing to change identity every so often. I'd probably stay in Europe, either in my home country of the UK or maybe even move to West Germany. Both countries did well in the post war 1950s, although not as well as the US. Maybe I'd go to the US because post war may have been an easier time to slip in to the country and then take part in the post-war boom?
I'd start trying to earn money, and put every penny I could into a company I owned and use that to buy stocks and shares in companies I know would do well in the post-war boom. The key is compounding investments to make more money longer term, accepting lower standards of living initially. In the 40s-60s that'd be largely blue chip companies like Coca-Cola, IBM, GE, Proctor & Gamble. Then as new big names of the future came along I'd buy into them - obvious ones being Microsoft and Apple in the 1970s, Google in the 2000s. And of course, probably buy into Warren Buffet's Berkshire Hatherway at the earliest opportunity in the 1960s.
With wealth I'd create new identities - a "son" to inherit the business every 20 years or so. Move around every 10 years or so, maybe between countries to help keep anonymity and to stay under the radar. Unfortunately it could also need a degree of criminality to be able to keep this going - identity fraud but at a low level. Maybe set up a trust, and every 10-20 years be a new lawyer who manages it. So much of that can be done somewhat anonymously and somewhat remotely even in the past.
The aim - if I am to be immortal - would be to accumulate a lot of wealth, spread across multiple countries, to prepare for getting back to today when fore-knowledge runs out. Because not knowing what happens next, all you can do is then try and prepare for possibilities, and making shit ton of money seems like a sensible way to guarantee security and comfort.
EDIT: The other thing to do of course: buy and sell land. Knowing that cities will grow rapidly is perfect for buying up land for later development. And also in many big cities, certainly in Europe anyway, there was a period of suburbanisation with matching collapse of the urban core. A lot of apartments and buildings were sold cheaply as demand was depressed, but then the trend started reversing again in the 1980s though to now. Imagine buying up apartment buildings in London, Paris, Berlin etc. Then all you have to do is live off the income - pay your taxes and no one really is going to ask who owns what, and who exactly is getting the money.