My approach is thus:
1. Search for patterns: does he come every 4 days, once every fortnight, from the west, similar TF make-up etc.
2. Is the bombardment part of an invasion armada or one offs (we already know it's the second in this case - but just doing the 'thinking').
3. Now, either follow jimh009's advice, with a SCTF on react, if you don't want to over-commit, but if you really want to give him a bloody nose try this:
Look for the pattern, try to find the time you'll get your highest likelihood of his arriving.
* Have a SCTF on react
* Have another SCTF, give it move orders to the base-in-danger, but with retire-allowed orders. This TF must be within cruise distance of your endangered base. This TF should move in at night and retire again by the morning (this can be a little difficult to keep secret, especially if they get low on fuel - leaving them at your base by sun-up, so maybe include a mid-ocean refueling option). This TF should be able to rinse and repeat as long as it has fuel.
* Put dedicated anti-naval planes at your base. These should be safe from damage as a naval engagement should cancel the enemy bombardment mission.
* And for the coup de grace, put a carrier force in range of his likely retreat track. Obviously this is a dangerous option and wont fit all situations.
If the moons are in alignment for you this should happen:
Your SCTF that moves in and out of the base will find the enemy bombardment TF;
The SCTF set to react, will react, and also meet the bombardment TF;
The enemy bombardment TF converts to a SCTF and does not bombard;
You damage enemy ships, and most importantly, use up their op points;
By sun up, damaged ships and ships that have used up op points that they cannot use for movement, are left within range of your fully operational battlestation (ahem - I mean your base) and it's anti-naval ACs, and hopefully you can finish off some of the cripples.
If nothing else he'll think twice about doing it, and it may lead to an escalation of commitment that he really doesn't want to make.
Image courtesy of Divepac