I wouldn't worry too much about a little clicking on startup, especially if it goes away in short order. This is most likely those hydraulic lifters pumping up. Allowed to do this at low speed, such as idle w/o the AC on, you're probably just experiencing normal wear. I wouldn't open the engine for a little lifter click.
Wait until the lifter click stops before raising engine speed or loading the engine. Some of this problem comes from increased clearances in ALL of the bearings, which means more oil is escaping via bottom end bearings. Add in oil pump wear, maybe a weak bypass spring, and it takes longer to get the oil up to the top of the engine.
Make sure the oil filter you use has the correct design. I mention this because after 1984, Toyota redesigned their oil filters to prevent oil from draining out of the filter after shutdown. The Supra motor was having oil starvation at the top end on startup because the pump had to first fill the oil filter before any oil would be pushed up to the cams. An anti-drainback vlave in the filter fixed that problem.
Just in case, try a Honda filter next time, and see if you have your valvetrain getting quieter, and sooner.