It lasts up to 14 days at room temperature (up to 25 Celsius, IIRC). Add in a chilled medication pouch (the sort that's used for insulin) and a letter from the hospital/GP to carry them in your hand luggage with a travel sharps bin and I don't see it being a problem.
The hospital will be able to give you more specific instructions, though.
I've remembered a couple of other things;
You might need to take a loading dose, which is having more injections than usual for a short time to get the levels up quickly - I think it's also a bit of a desensitisation to doing the injections as you get used to the process within a few weeks,
And before you start it, you'll have a few tests - chest x-ray, one for TB if you don't have a BCG vaccination and some more blood tests to include HIV and Hepatitis (I think), as taking biologics can activate dormant infections. I think there was a chickenpox immunity as well - if you come into contact with chickenpox, the procedure is to call your GP for an urgent prescription of antivirals (which are fine, they don't taste bad or have any side effects in my experience).
You definitely can't go places that require live vaccines like Yellow Fever and if you were harbouring dreams of becoming a bat conservation volunteer, you can't, as that requires Lassavirus/Rabies vaccinations (that was an interesting question for the doctor - 'Why would you want to handle bats?').
And look after your feet. Treat them as though you are diabetic, wearing footwear all the time instead of risking cuts, glass or other things in the kitchen, making sure you don't get blisters or ingrowing toenails, much as you might have been told to do with MTX.
It's likely you'll develop the ability to not flinch at multiple vaccinations and be able to tell the phlebotomist before they start exactly what vein to use and how to get it nice and plump for blood tests - 'Uh, not that one, use this one (gentle stroke along the vein as it pops up).'
It sounds a massive thing beforehand, but it's 99% insignificant with a few minutes a month just getting on with it and 1% of you starting conversations with 'I'm on Biologics for...' as that focuses attention for all infections, illnesses and GP appointments often become available when there weren't any beforehand (if you don't already experience this with MTX).
I'd probably ask for a steroid pulse to get you started, as coming off MTX can mean a flare before your biologic blood levels get up.
I've not been noticeably susceptible to infection since starting, but your mileage may vary. I think biologics are great, though. Because when you get the right one, it really, really works.