For playback the best codec is unarguably h.264 (AKA MPEG-4 Part 10, or MPEG-4 AVC, or just plain 'mp4'). H.264 in an mp4 container allows for small files with good visual quality, the ability to tune the settings to achieve tradeoffs between size, compatibilty and quality, and almost universal playability with modern devices.
In the Premiere export window choose H.264 as the format. There are numerous presets available, but most won't be relevant because of the unusual dimensions of your video. The ones you should be looking at are the ones called Match Source-[...] bitrate. Depending on the playback machine you should try the High bitrate first. Adaptive High Bitrate will give you a lower file size by increasing the compression (= lowering the quality) in parts of the video that don't need it, but for local playback this is not really an advantage.
The Match Source bit means that the encode will be done at the same frame size as your sequence.
If you are having trouble playing back the mp4 you might want to look at the profile in the Video tab of the encoding settings. Simply put, this determines how comlex the task of de-encoding the file is. For lower-powered or older machines use the Main profile, but if your playback machine can support it use High Profile.
High10 lets you use 10-bit encoding. For a final playback file this is usually unnecessary, unless you're using a 10-bit monitor (if you don't know then you're almost certainly using an 8-bit monitor) it won't make any perceivable difference, and will limit the compatibilty for playback.