Probably Rare, But Difficult to Rate Based on Abilities

[Note, at the time of writing this the item seems to stop any sort of damage 10% of the time, but I believe based on past variations that it may be the designer's intent to stop only melee attack damage. In the latter case I would rate it uncommon, as it seems, on balance, to fall a little short of other items mentioned. Please leave a comment if the item description is changed to clarify this so that I know to edit].

There are two damage mitigating abilities here, so let us consider both of them separately.

First we have +1 AC under certain circumstances. A straight +1 armor is a rare item which doesn't require attunement. A Cloak of Protection is an uncommon item that does require attunement, which always applies and also gives +1 to saving throws. The AC on certain advantaged melee attacks (which would cover flanking for those tables with that variant rule) is far weaker than a straight +1 to AC. A far weaker version of an uncommon item's ability should not increase overall rarity by too much.

The other ability is much trickier. 10% of the time when the wearer takes damage it results in damage immunity, and generally damage immunity is a very high level ability in any form (immunity to only one or two types of damage is the providence of the legendary Efreeti Chain armor, or a 18th level Storm Sorcerer ability). However with this item the immunity comes comparatively rarely and unreliably, at unpredictable times, and there is a 1% chance of double damage, which makes the calculus of having, using, or relying on the ability very different.

I think we have to look more broadly at damage mitigation. Avoiding damage entirely comes in the game mostly by way of increasing AC and spell save. Ultimately the average damage avoided will be substantially less than +1 to AC and spell save would, since that saves the character from damage more than the net 9% of the time this will. Once again this has your cloak coming up relatively weak versus various uncommon and rare items (Cloak or Protection, Ring of Protection, +1 Armor, etc).

It is true that, unlike any of these items, this item has a ten percent chance of saving you even from a natural 20 crit, but it also has a one percent chance of doubling that crit, which at lower levels could be very deadly indeed. It also can protect you a small percentage of the time from all other forms of damage such as area attacks that do half damage on a save, spells and abilities with guaranteed to hit attacks (eg: magic missile), or potentially most dramatically impact damage from falling off an airship, being crushed by an avalanche, etc., etc.

Ultimately I would say that, combined, these abilities, along with filling a relatively common magic item slot and requiring attunement, leave the uncommon Cloak or Protection as the closest analogue in terms of practical effects and the nearly identical, rare Ring of Protection as the next closest. In terms of preventing the sort of damage these items will it comes up a bit short. It seems, however, that it can provide universal damage immunity potential, albeit in a highly unreliable way. This definitely tips the scale towards rare, but I would not be surprised if playtesting proved it more in line with a very rare item.

The ability to occasionally protect a character from any sort of damage also seems like something that would make it desirable to many players, perhaps beyond its actual practical value, something you may want to take into account.

Likely Rare

The first benefit is an extremely situational version of the Cloak of Protection (minus the +1 to saving throws), an Uncommon item. Uncommon items are considered +1 items. However with the significant restriction that they attacker must have advantage, that you must not have just attacked them and the loss of the +1 to saves this ability is really only a Common. Hence unlikely to increase the rarity significantly.

The second benefit we have no precedent for, negating damage like that isn't something that commonly happens in 5e. If we were to consider it equivalent to imposing disadvantage on the attack roll* (a more sensible ability) then it would be the same as the Cloak of Displacement, a Rare item. Rare items are +2. The Cloak of Displacement has a limitation to only work until it is hit each round. This offsets some of the differences between disadvantage and 10% miss chance.

This damage reduction could also be considered equal to a +2 to AC. Increase AC by 2 also reduces incoming damage by 10% (sort of). +2 Armour is a Rare item. So either way this ability is roughly equivalent to a rare item.

The 1% chance of doubling damage is fairly insignificant and balances out with the first benefit. So in total this item is mostly likely Rare.

*10% Miss is slightly better at reducing damage at low AC < 10 and gets worse as AC increases. See this anydice calculation for the math.