Yes, this will work fine
Cat5 is rated for Gigabit Ethernet (1Gbps) for 100 meters / 328 feet. (source: IEEE 802.3ab). Cat5e has more crosstalk resistance than Cat5 and is the most inexpensive network cable available today. It will be more than sufficient for your needs. Cat6 is even better and will work fine too.
Another poster said that you need to put in a wall jack. I agree that this is the best way to do the job. However, it sounds like you are more interested in getting the job done without it being "pretty." A 300ft patch cable should work just fine, and will save you the time and expense of purchasing a crimper and learning how to terminate.
A few notes:
- Triple-check your measurement before ordering the cable. I would add ~20ft to be safe.
- Don't let the cable touch the ground under the house. Attach it every 5-10ft to a beam with a cable tie or equivalent.
- Don't run the cable parallel with electrical lines that are not in conduit. This will hurt the performance of the link.
You do not need to worry about 300ft of Cat5e or Cat6 having performance problems, particularly not with a 9Mbps internet connection.