Please be careful of the adamoh.org site - Gunnar get's terribly muddled and frequently his info is incorrect.
I have a wonderful free program called Stellarium that I use out in the field to locate and identify objects during my observing sessions.
Stellarium is a lovely program for more modern skies. IMO it isn't that great for ancient skies. The moon is often shifted a few degrees further to the west compared with other programs I've used, like SkyViewCafe and Alcyone Ephemeris. The program, I think, fares better with planetary and stellar locations, however.
First I check April of 588 BCE. No good. Saturn is in Gemini. Way off.
Please bear in mind that, according to Furuli, the planetary positions on the tablet fit 568/7 BCE but he thinks they were likely retro-calculated centuries later (by the Seleucids) to fit with a revised chronology which mixed and matched 568/7 planetary data with 588/7 lunar data onto one tablet. The WT article just claims the planetary names or positions on the tablet are too ambiguous or unclear to be of any use. Of course, as has been pointed out by Doug Mason and Carl Jonsson in their Critiques, this claim is pure nonsense.
Just to emphasize: Furuli and the WT article claim that only the 13 lunar positions relative to stars are genuine observations belonging to 588/7 BCE. All other positions on the tablet - the lunar positions relative to the sun (i.e. the 'lunar three' time intervals) and the planetary positions - are disregarded.
Here is my star chart.
Just a small problem - even though Saturn is approaching Pisces - I notice your date is -568-04-23. The astronomical date for 568 BCE is -567. This is Saturn's position at the beginning of Nisanu in 568 BCE: