If you can remember the settings for your old network (SSID/network name, 2.4 vs 5GHz, and password), configure your new router to match and it should connect. Otherwise you need a remote. There's no way to use the app until it's connected to your network.
I'm not sure what you mean by "turn on". Roku devices (other than the Roku 4 and Roku TVs) are always on. Is there any light on the front of it when you connect the power? What's the model number? It should be written on it somewhere.
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