There is no rules exception for same-round casting of spells from magic items.
In the DMG, under "Activating Magic Items", it says...
Some magic items allow the user to cast a spell from the item, often by expending charges from it. The spell is cast at the lowest possible spell level, doesn’t expend any of the user’s spell slots, and requires no components unless the item’s description says otherwise. The spell uses its normal casting time, range, and duration, and the user of the item must concentrate if the spell requires concentration. Certain items make exceptions to these rules, changing the casting time, duration, or other parts of a spell.
Other than the fact that they do not require components or slots (first bold section above), spells cast from items are no different from casting them normally... unless the magic item specifically calls it out (second bold section).
The Necklace of Prayer Beads specifies a reduced casting time for the spells it contains, but that's it. If it were meant to bypass any of the other spellcasting rules, it would say so.