Can we simulate FPGA board?
If you need real interaction at real speed, then no there is no alternative. You can get quite creative with the testbenches, but if you need to send to a display then you're not going to be able to do that in software.

In some ways, FPGA is the simulator you're asking for; it's a simulation of an ASIC.

(If it's part of the curriculum it really ought to be provided by the course! Usually as a loan.)