Fresh boiled syrup is practically sterile. Nothing can live for long at those high temps. Anything that leads to spoilage in your syrup will be introduced in the bottling process so keeping all equipment extremely clean is imperative. That includes, ladles, funnels, whatever.
Sterilizing jars/lids won't hurt but may not be necessary. No one is sterilizing those plastic jugs or bottle caps that a lot of syrup gets sold in. With syrup, the enemy is yeast and molds.Those are killed at temps in the 150's. If you start with really clean containers, keep your equipment clean, pack at the right density and at the right packing temp, and fill the bottles to capacity, you should get good seals and syrup that will last without spoiling.