Box Mode in 1v1. LOL is quite a small mode in terms of area size. A two-on-two experience, Box Mode allows you to build defences as with the other modes. You and your teammate can do this to create the best defence for yourselves in each match.
But, with such a small arena to work with, it might be best to go small with your forts. There's no need to go big here. Also, be careful not to trap yourself when building. You'll likely already see your teammate build things as well, so you can limit things.
So it's really all about the space that's being used. If you want to, you don't need to build in this mode and can just try and go for the kill. It might be kind of tricky if the opponents create their own forts, but you might be able to get in there and sneak in some shots.
In such a small area in Box Mode in 1v1. LOL, it's a smart idea to probably stay on the move. If you aren't a player who relies heavily on building defences, then you can keep moving around. In all honesty, you probably are doing this already. It's pretty natural.
When you have a fast-paced, two-on-two match, it's easy to be on the move. It's more of a defensive move, but it can also be a solid way to play a bit of offence. If you feel like you can shoot and run, then you might be in a good shape. Just know that opponents may do the same.
But, since you have a teammate, they can possibly bail you out if you feel like you're in danger. Just make sure their teammate isn't nearby. Box Mode is pretty chaotic for a duos match. The enclosed arena and how fast players can get killed make things hectic.
Feel Free to Use a Shotgun
With 1v1. LOL having most of its matches played in close proximity, this gives you some options in regards to your weapons. Well, you have two gun choices. One is a machine gun and the other being a shotgun, both available from the jump.
You can choose whichever weapon you please (heck, you can even use the axe if you like), but as a reminder, the shotgun is a solid weapon for close-up battles. And as we've been talking about, Box matches are in tight spots, hence the name, because you are essentially in a little cube.
You can deal some good damage when in personal range. However, if the fire rate of the shotgun isn't your fancy, you can always go to the machine gun which is fun to use. But definitely feel free to experiment and see which gun works best for you.