You have several concerns and the limiting factor here is money. We'll have to assume you're a handy person and can do your own installations.
You want to buy a security screen door for each exterior door of the house (or at least one for the front and one for the rear and you can seal off side doors). These will run you about $300 each, but you can sometimes find them used, especially if you don't care if they are an exact fit (as long as no zombie hands can reach through, you're okay) and aesthetics.
Window bars aren't that expensive, but they will add up fast, depending on how many windows you have. This 3x4 foot one is $34. Again, they're often available used and do check your local junkyard or recycling place. Exact fit is not important.
Your fence is going to be the more expensive issue. It's in good shape but privacy fences are easy to climb and to break through. It depends a lot on how resourceful, agile, and desperate the zombies are. Can they get and use ladders? Move large objects to the edge of your fence and use them to get over? Can they just stand on top of each other?
They may not bother someone with a fence in the beginning, but as they start to die off and their lack of food makes them more desperate, they are likely to take more extreme measures to feed.
Securing the entire backyard is not possible on your budget (and timeline). Instead, put your resources into creating a cage out of chain link fencing (or similar fencing used in zoos and kennels) around your back porch/patio area only. Don't forget to fence in the roof of this extension.
This will protect your washing water that doesn't fit in the house, other large supplies, and your cooking area (and your toilet area if it's needed).
Note that your security is to 1) keep out zombies, and 2) keep out other humans.
This is easy. Go to Grocery Outlet or some other discount place and get lots of shelf stable food that doesn't need cooking. Cans of beef stew, tunafish, etc. Things that are cheap and high in protein and calories. Consider also restaurant supply places. The prices are super cheap for a lot of things and most of them let you shop once without a membership (at my local one, membership is free but you have to have a business tax ID or other proof of a business, which does not have to be a restaurant). Make sure you don't buy anything that is perishable after being opened if you can't finish it before it goes off.
Buy bags of beans and rice and dried peas and lentils. Fresh vegetables that easily keep a couple months without refrigeration: carrots, potatoes, onions. Dried fruit, jam, etc. Give yourself a variety. Stock up on cooking oil and herbs/spices. Make sure there is plenty of salt.
You don't want to starve and you don't want to be tempted to leave the house. Buying in bulk is fairly cheap and there are places that sell stuff cheap because the labels got damaged or the manufacturer changed the packaging design.
Camping stoves are cheap (a heavy-duty single burner one that connects to large propane tanks is about $38), these are available used or sometimes in thrift stores (test them!), and run on propane. An empty 40lb tank runs about less than 100 bucks and holds just over 9 gallons of propane. Fill it up for 18-36 bucks. Buy several tanks.
The above stove with the above tank will last for about 15 hours of cooking. But that stove is very strong (55,000 BTU) and is suitable for boiling large pots of water, making pots of beans, etc. Have a smaller camping stove as well for tasks like heating up small amounts of water and food (this double burner one runs $43). These only use more like 10,000 BTU per burner. So your tank will last for about 135 hours.
If you already own a charcoal grill, you can use that too, but the smell will broadcast your location (especially to hungry and desperate humans) and it must be used in the backyard, which may not be possible after a while.
Buy a big aluminum pot for boiling water and etc. Thrift stores and garage sales will have these plus camping stoves for cheap.
Drinking water will be okay after a few weeks in opened containers, but buy purifying drops just in case. If your water is chlorinated, it should be okay to put tap water into old water bottles and close them. You can also buy sealed bottled water, but it tends to run about $1/gallon.
For handwashing water, use tap water to fill up large empty drinking water containers with spigots. Enlarge the vent hole on top until you can get a funnel in there. I do this for camping and it works great for hands and dishes.
The standard is one gallon per person per day. Two people for 4 months is 240 gallons. About half of that should be sealed drinking water. The rest can be potable tap water in clean closed containers.
Find used wading pools that do not have any leaks and fill them all in your yard with the hose. This is your water for bathing, cleaning, dishes, and sanitation. You can do a final rinse of dishes and stuff with drinking water if needed. Have buckets of various sizes.
Your toilets will still function!! The only thing that will be missing is the incoming water to fill the tank. And it might not even be turned off. If it is, know that all you need to flush your toilet is 2 gallons of water quickly poured into the bowl. You can fill the tank itself but just pouring into the bowl is probably more efficient. Test this out before the zombies come because you want to know which works better for each of your actual toilets (they're all a bit different).
As a backup, get two chamber pots (so you and your wife don't have to share). And dig a hole in the backyard and cover with plywood. You can have a squat toilet or just use the chamber pots and empty them. Have a smaller hole nearer the house (in your outdoor security area) if you can no longer use the backyard. Make sure some of the washing water is reachable (fill your bathtub and also containers in the house).
You can stay clean enough with bowls and washcloths but also consider a solar shower. They run about $20.
Buy a case of two of toilet paper at Costco. Have plenty of dish soap, hand soap, etc. Cheap clean towels. A clothesline outdoors and one indoors. Stoppers for your sinks to use them for washing (the drains will still function). Don't get behind on laundry before the start of this mess. Keep dishes washed and don't let them pile up.
Yes, you can have enough food, water, and sanitation with your budget and setup to last you 4 months.
The issue you might not have the money (or prep time) for is security. But if you plan that you may lose access to your backyard and just secure the doors and windows of the house itself, you should be okay. Ideally you should have a back patio area secured as well. You don't have the money (or time) to secure your entire backyard but you should have access to it for a short time at least. It is best to do your cooking and toilet needs (if the house toilets stop draining) outdoors and water storage is better outdoors as well due to space. But you can do it all inside if you have to.