It's a tricky one. You can't get it 100% correct with AR's, let alone without. I've refereed in areas with and without CAR's - I'll take refereeing by myself over CAR's anyday, and I haven't even had a particularly bad CAR!
First off, reconsider your positioning. Positioning is about considering the various events that might happen, how likely Event A is compared to Event B, what's the relative significance of Even A compared to event B (A might be more likely, but B might be a game-changer), and what are you sacrificing for your position. One position might give you a great view of A and no view of B, the other gives you a moderate view of both.
With a NAR, you can take offside position judgement out of that equation almost entirely. Offside may be game changing, it's quite likely to occur, but everywhere you have a near-perfect view (because of your AR), so position yourself for other things instead.
Without an AR, you need to reconsider your positioning. Diagonals may become pointless.
Take a ceremonial FK, for instance. Normally you're sort of in front and to the side, looking into the PA at an angle with a view of the wall. But that gives me zero view of a close offside - which is fairly likely to occur. So I'll basically adopt the AR's position - on the field, but directly in line with the 2nd last defender. It compromises my position to see everything else, but I've still got some view (and don't forget, the 'traditional' position can still leave me unsighted for events that may occur behind the wall - no position gives you a view of everything that may occur). That position also allows me to quickly move to the goal line if there's a shot as well.
I've had games with one NAR, and I've found myself positioning almost where other AR would normally be - spending most of the game up near the other team's sweeper. This has just been the occasional weird game where there were few fouls, even challenges in the middle of the park, lots of long balls....basically, really weird positioning, but in that occasional game I don't believe I missed anything. Just an example of how you reconsider your positioning.
As for spotting offsides - there's a few things you can do to help. First off, you need to preempt play more. Constantly glance up to where the attackers are. Who's looking to start a run? Who might be free? Where is the ball carrier starting to look? Who's calling for the ball? But you want to do this without looking away from the ball-carrier - so sometimes you may need to get wider or deeper to have the offside line in your peripheral vision. You may need to start running upfield a bit earlier as well
Without an AR, I'll sometimes find myself standing off the field for throw-ins as well - after all, you also need to adjust your positioning for ball in/out of play. I change my corner position to standing off the goal line, around the GA, as opposed to infield around the corner of the PA.
Back to offsides, use the field markings. Not just the PA, but you might have a slightly discoloured line of grass across the field where there's drainage or something. I also find that if I quickly run my eye back and forth along the grass, almost like I'm tracking an imaginary line, then I can almost draw a virtual line in my mind. This has helped me a lot. Try it on a field. Stand in the middle of the field, pick a spot near the touch line, say, 5-10 yards out from the PA, quickly run your eye along the ground to the other line. You should have an idea of what bits of grass were on that, compare them to the PA, and you'll see it was pretty accurate. I found myself starting to do that in play - I may not be sure if somebody is in an offside position, but I 'draw' my imaginary line and I'd feel like my view is just as good as if I had an AR.
Finally, reconsider your position during the game. If you feel like you might be unsure of a few close ones, then change your positioning to pick them up. Sometimes it's just hard to spot - especially if the attacker is usually a long way across the field from the nearest defender, these are very tricky. I've had games where the red team's balls going through, I'd even be able to spot the close offside but for the red team, I'd struggle to have a clue just because of the relative positioning of the players. It's tough, and offside will cause problems.
And sometimes you'll incorrectly call an attacker offside as well. It's really the fault of the players running a close offside line without an AR!
