Can I replace two 15 amp fuses with 10 and 20 amp fuses?
Nope, either way

First, "upgrading" the 15A fuse is not allowed.   A few people are talking about the rules by which the circuit might've been wired for 20A.  That's dreaming.  If the circuit met those requirements, they would've used a 20A fuse in the first place and you wouldn't be having this discussion now.  So I wouldn't go getting my hopes up on that.  

Now, it's not illegal to swap the 15 for a 10. However, it's a bad idea because then, the next guy won't know what fuse is supposed to go there.   You may notice something about that fuse box, there is absolutely nothing on the box that tells you what the fuse should be. 

That 30A rating is the rating of the fuse box itself.  It is saying that you could hypothetically use that fuse box with dual 30A fuses, 10AWG wire and feed an electric dryer.  However in your case, the wire is almost certainly #14 AWG, which only allows 15 amps.  

I would be surprised if the refrigerator were the "other" appliance.  They tend to be quite low draw unless they are older units, and if they are older units, most electric companies will be happy to give you a new fridge, because it's cheaper than adding generating  capacity to power your old one... and I would be amazed if your landlord would oppose a free new fridge.  He probably just hasn't bothered to ask because he doesn't pay the electric bill and doesn't have time to deal with it.  This is an opportunity for you. 

Now it may be a different appliance, perhaps one that you consider inconsequential, because most people frankly don't know what devices draw.  They think "$20 toaster must draw a tiny fraction of the huge expensive fridge" nope, it's the exact opposite.  If you can identify that other appliance, that may solve it.  

Most kitchen appliances which use heat use 1500 watts (12.5 amps) whether large or small.  It wouldn't matter if you had a brand new house with fresh 20A wiring, no kitchen circuit can support two 1500W (12.5 amp) appliances at once.  

It's also possible you have a large deluxe microwave that pulls 1500+ watts and cannot share a 15A circuit.  Take it back and get a 1000W mini.