It is not uncommon for employees to pay for travel on their personal cards and then get reimbursed. I'd do a couple things.

First, talk to your employer and make sure you get a commitment as to the reimbursement timeframe, as well as what is needed to process the reimbursement (e.g., all receipts or just receipts over a certain amount, what's covered and not, meal limits, etc.). You don't want to get into a situation where it takes your employer a month to pay you back, or where several of your submissions are not considered valid after the fact. Also, if they can accommodate it, a direct deposit setup can be nice so you're not dealing with physical checks. The idea of a cash advance is also a good one, if you can estimate out certain expenses (hotel + taxes/fees, average meals, ground transportation, etc.).

Second, if you need to, talk to your credit card issuer and get your limit raised (if necessary) and inform them that you may be making some large travel-related purchases (if out of the ordinary - sometimes there will be holds placed if you make large atypical purchases, like out of your home area).