Eluned Morgan said red category life-threatening calls would be prioritised but people "need to consider" pressure on other calls "will be even greater".
The Welsh Ambulance Service said it had been experiencing high call demand as a result of the icy weather.
It has called on people to only dial 999 in a life-threatening emergency.
The UK government has made arrangements to deploy 1,200 members of the military and 1,000 civil servants in England during Wednesday's strike.
Ms Morgan said the army had a role to play but soldiers would not be driving ambulances in Wales.
"We will not be using the army to transport people, but we are going to be using the army for logistics," she told BBC Radio Wales' Sunday Supplement.
Asked if she was confident people in with red category calls would get an ambulance during the strike, Ms Morgan said: "We're hopeful they will if they are in that category on Wednesday.
"But that means the pressure on those not in that category will be even greater, that's something people need to consider.
"And, obviously, it's important that people seek clinical advice."
She said people calling the NHS Wales 111 advice line and the 999 service would be directed to the most appropriate help.
"And there will be occasions when you'll be encouraged to try and get yourself to hospital if it is genuinely difficult and not that life-threatening situation," she said.
The Welsh Ambulance Service recorded its worst-ever performance in October, latest figures have shown, responding to less than half of red category calls within eight minutes.
Some 10,000 ambulance staff are striking across most of Wales and England on 21 December in a dispute over pay.
Meanwhile, Helen Whyley, the Royal College of Nursing (RCN) director in Wales, said there will "absolutely" be more nursing strikes in the new year.
Welsh Conservative health spokesman Russell George called it "totally disingenuous" for the first minister to say he could do nothing.
"I am also surprised that the health minister is so candid, only being hopeful that ambulances will actually reach the most serious situations," Mr George said.
"However, that seems to be the case even when there isn't a strike on, with Wales recording its slowest response times on record in October.
"Labour need to get a grip on the NHS."