How to play League of Legends between 2 regions?

I'm going to be in Vietnam for the next few days while my friend is in Singapore. I recall the last time I was in Vietnam I was forced to download a different version of Garena and/or League of Legends.

Is there anyway I can still play with my friend using the Singapore server?

The most important thing here would be the server you're going to play on. You both have to play on the same server (region) to be able to play with or against each other. For example, you both have to play on the Vietnam (or Singapore) server.

It's important to note that, due to the way the internet "works" in that region of the world (and due to different contracts), you both might be forced to play on different regions based on your IP (as you already mentioned from your previous experience). The only way around this might be using VPN software, but this might cause additional trouble (additional lag, being suspicious, etc.).

Overall, it's really hard to estimate this, you'll probably have to use trial & error for this (even with IP restrictions you might not necessarily be restricted based on your actual IP sub net for example).

Assuming you mean 2 different servers, you can't

You'll either have to transfer if that's possible on your server or create a new account on the Singapore server.