Can an LPC1343 accept its initial program over USB?
Details of USB booting are provided in the User Manual on page 294:

The LPC134x is enumerated as a Mass Storage Class (MSC) device to a PC or another
embedded system. In order to connect via the USB interface, the LPC134x must use the
external crystal at a frequency of 12 MHz. The MSC device presents an easy integration
with the PC’s Windows operating system. The LPC134x flash memory space is
represented as a drive in the host file system. The entire available user flash is mapped to
a file of the size of the LPC134x flash in the host’s folder with the default name
‘firmware.bin’. The ‘firmware.bin’ file can be deleted and a new file can be copied into the
directory, thereby updating the user code in flash. Note that the filename of the new flash
image file is not important. After a reset or a power cycle, the new file is visible in the
host’s file system under it’s default name ‘firmware.bin.

Page 291 describes how to connect the device pins to ensure that USB booting takes place.