The main circuit breaker on the side, at the outside out my house is
overheating and trips. All electricity inside the house gets turned
off when this hapens. It tends to happen whenever I use the AC on a
hot day. The electrician says that this 90 AMP main circuit breaker
needs to be replaced. The problem is that he says that they don't
make these 90 AMP circuit breakers anymore. They only have them in
100 AMP. He says it's safe to go up from 90 to 100 AMP.

Is this safe to do at my house? Your suggestions/advice would be
greatly appreciated.

I'd say go for the 100 amp brkr. It's a good start point, lots cheaper
than any of the alternatives, and sounds like sound advice from someone
on site that could have tried to sell you a bunch of expensive fixes,
assuming his license and insurance are current.
Be sure he gets the required permits and permissions if needed.
He'll know what they are but you can also always call your local zoning
office for info too.

If that doesn't fix it, then perhaps more expensive repair will be
needed, but IMO it's a very sensible place to start.
The over-price will be negligible if more serious work is required,
but I don't think it will be.

You also *could* have someone make measurements (unless it's already
been done) to insure that a new breaker is going to suffice; but if it's
a 90A, it's fairly old already to start with, if it's outdoors,
conditions are worse on it, etc. etc. etc.. Measurements will cost
another few $ but sometimes it's better to be sure right up front when
you're close as you seem to be. It's quick & easy to make such
measurements though you might have to go in and turn a few things on
while they're measuring.
Appreciate your electrician; he seems to be doing the right things to
get you out of trouble at minimum expense.

HTH