First, you will need to gain access to Ichigaya to unlock fishing. You can do this by reading the book, Fishpond Spotter, or by accepting Ryuji's invitation to hang out. Next, you'll want to read the Essence of Fishing book, which can be purchased in Shinjuku after you've fished once. This will make fishing significantly easier, as you'll be able to see the fish better with Third Eye and see where your line lands.
Once you've done this, you will need to devote time to fishing to accumulate enough points to purchase Top-class Boilie (400 Pts), which is needed for the Guardian. You can also purchase better fishing rods, but this is not required as they just increase the number of times you can maintain your line before a fish gets away.
To get these points in a timely fashion, you will want to catch fish with tags that are worth double points. You can tell if a fish is tagged if they sparkle when using Third Eye. Fish are more likely to be tagged when the weather is Raining and less likely to be tagged when the weather is Cloudy, so pick your days wisely.
It also may help to purchase Medium and Large Boilie baits to catch bigger fish for better point returns.
Now, onto the actual method once you have everything you need. A Guardian will spawn randomly, but frequently. It can show up after you've caught about 6 things in any weather, but it can show up earlier if you go when it is Raining. You'll know when one appears when you see a long, gold fish with Third Eye.
Catching him through conventional means is difficult to do, even with the best rod, so instead you'll want to lure it to as close to you with bait. The trick is to throw your bait in front of it, and after it begins to move towards it, pull back your bait. This won't consume your bait or the number of times you can fish, so you will be able to throw it again to repeat the process until it is as far south as possible.
Note: Unlike other baits, Top-class Boilie attracts all fish, so you'll need to retract before another fish gets too close.
Once you've lured it close enough, let it latch onto the bait, and you should only need a few button presses to actually get it.
All credit to xNinjaOfChaosx