Possible to use FPGA to emulate TTL logic ICs in existing circuit?
In theory: definitely yes.
In practice: Yes, in some cases, but you won't use a Cyclone IV for this.

First, you have to make sure that the FPGA is electrically compatible with the board. That is, you must be able to set its I/O voltage to the voltage required by the board.
Then, the driving strength has to match - FPGA usually don't deliver more than 12 mA per pin, while many 74xx chips offer a lot more. 
Then, the I/O type has to match - is the broken 74xx TTL or CMOS? Does it have a push-pull output stage or e.g. an open drain output? Does it have pull resistors or a hysteresis on the inputs?
What are the requirements for propagation delays of the chip? Rise time of signals?

For common chips you will be able to find the necessary settings in an FPGA, but for special ones it will be impossible. Given that you have an FPGA that can deliver the required electrical performance, the huge drawback of large FPGA like your Cyclone IV is that they need a large amount of external components (Flash, voltage regulators...) and that is very impractical to provide "just" to replace a 74xx chip.

On the other hand, there are several tiny FPGA on the market that only need a single supply voltage and no further external components. Examples are the ice and MachXO series from Lattice, sizes start at around 3x3mm² and prices are about 1 $€£. So, you might actually be able to fit one into the socket of the original broken 74xx chip.

Summarizing: Yes, it might work, but it's an awful lot of work and not a viable way to go in most cases.

Some examples of what can not be done with FPGA:


All legacy 7400 from the original series can not be emulated without external level shifters, because they use 5V I/O not provided by any current FPGA
Some 74xx series, like 74CBTLVxx offer very fast propagation delays of less than 2 ns. This is faster than most FPGA that have a pin-to-pin delay of several nanoseconds.
There are devices like 74xx19 with a Schmitt-triggered input. These have a large hysteresis that can't be provided in FPGA without external components.
There are devices like 74xx06 with an high-voltage open-collector output, far beyond the capabilities of an FPGA.
There are analog chips like 74xx624 that is an voltage controlled oscillator - that's outside the domain of what an FPGA can do.
There are high-current bi-directional analog multiplexers (e.g. 74CBTLV3257) - you could replace these with some transistors, but not with an FPGA.