You cannot cast two leveled spells per turn that way

The ring of spell storing says:

The spell uses the slot level, spell save DC, spell Attack bonus, and Spellcasting Ability of the original caster, but is otherwise treated as if you cast the spell.

Nothing in the description of the ring does it say that an action must be used to cast a spell from the ring. Therefore, "otherwise treated as if you cast the spell" suggests that you need to use an action, bonus action or reaction as appropriate, to cast the spell. This is strictly determined by the requirements of the spell to be cast.

The rules on spells cast as bonus actions state however, that in addition to such a spell, you can only cast a cantrip having a casting time of one action. You cannot, therefore, cast sanctuary (one bonus action) and and beacon of hope (1 action, not a cantrip) on the same turn. This is a result of the rules for spellcasting, however, nothing caused by the way the spells are cast from the ring.

You cannot cast a levelled action spell and a levelled bonus action spell in the same turn.¹

The type of action required to use the ring of spell storing is determined by the timing of the spell being cast. From the Dungeon Master's Guide on activating an item:

Some magic items allow the user to cast a spell from the item. 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.

In this case, beacon of hope has a casting time of 1 Action, and sanctuary has a casting time of 1 Bonus Action.

You cannot do this, the rules for bonus action spells state:

A spell cast with a bonus action is especially swift. You must use a bonus action on your turn to cast the spell, provided that you haven't already taken a bonus action this turn. You can't cast another spell during the same turn, except for a cantrip with a casting time of 1 action.

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¹Here, "levelled" means "having a spell level of 1st or higher".