There are a ton of apps for this. I personally used one called Stock Trainer, mostly because that was the exact search I put into the App Store. The problem I found is that games like these are really only for mimicking day trading. If you buy and sit, you're gonna sit for a long time. Like years. I bet you don't want to wait years for a game to confirm your investment ideas. You should also be researching what options are available to you. Unless you can really pick stocks (which the game will tell you...in a year or two), your best bet for a buy and sit is probably an index. Minimal effort, and the best average return (generally said to be around 12% per year, but this is the stock market).

There is also an online game that has you track individual stocks on a day by day basis using old charts. The goal is simple: best buy and hold. If anything, the game proves that for the average Joe, but and hold us the best strategy. I forget what the game is called, though it is linked from enough online resources you should come across it either in your normal research or through a quick search for 'stock game's or 'stock trainer'. I believe I found it through Investopedia.

Best of luck, and do your research.